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Properties

Label 8-126e4-1.1-c9e4-0-0
Degree $8$
Conductor $252047376$
Sign $1$
Analytic cond. $1.77350\times 10^{7}$
Root an. cond. $8.05571$
Motivic weight $9$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 32·2-s + 256·4-s − 961·5-s − 1.45e4·7-s + 8.19e3·8-s + 3.07e4·10-s + 1.53e4·11-s − 4.52e4·13-s + 4.66e5·14-s − 2.62e5·16-s − 1.42e5·17-s − 5.25e5·19-s − 2.46e5·20-s − 4.91e5·22-s + 1.79e6·23-s + 2.79e6·25-s + 1.44e6·26-s − 3.73e6·28-s + 8.48e5·29-s + 7.17e6·31-s + 2.09e6·32-s + 4.54e6·34-s + 1.40e7·35-s − 9.25e6·37-s + 1.68e7·38-s − 7.87e6·40-s + 5.38e7·41-s + ⋯
L(s)  = 1  − 1.41·2-s + 1/2·4-s − 0.687·5-s − 2.29·7-s + 0.707·8-s + 0.972·10-s + 0.316·11-s − 0.439·13-s + 3.24·14-s − 16-s − 0.412·17-s − 0.925·19-s − 0.343·20-s − 0.447·22-s + 1.33·23-s + 1.43·25-s + 0.621·26-s − 1.14·28-s + 0.222·29-s + 1.39·31-s + 0.353·32-s + 0.583·34-s + 1.57·35-s − 0.811·37-s + 1.30·38-s − 0.486·40-s + 2.97·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{8} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(1.77350\times 10^{7}\)
Root analytic conductor: \(8.05571\)
Motivic weight: \(9\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{8} \cdot 7^{4} ,\ ( \ : 9/2, 9/2, 9/2, 9/2 ),\ 1 )\)

Particular Values

\(L(5)\) \(\approx\) \(0.03533138545\)
\(L(\frac12)\) \(\approx\) \(0.03533138545\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( ( 1 + p^{4} T + p^{8} T^{2} )^{2} \)
3 \( 1 \)
7$C_2^2$ \( 1 + 2082 p T + 327613 p^{3} T^{2} + 2082 p^{10} T^{3} + p^{18} T^{4} \)
good5$D_4\times C_2$ \( 1 + 961 T - 1873839 T^{2} - 213128658 p T^{3} + 88270950034 p^{2} T^{4} - 213128658 p^{10} T^{5} - 1873839 p^{18} T^{6} + 961 p^{27} T^{7} + p^{36} T^{8} \)
11$D_4\times C_2$ \( 1 - 15361 T - 3701568735 T^{2} + 11956485133686 T^{3} + 9387835659282450844 T^{4} + 11956485133686 p^{9} T^{5} - 3701568735 p^{18} T^{6} - 15361 p^{27} T^{7} + p^{36} T^{8} \)
13$D_{4}$ \( ( 1 + 22639 T - 4548571174 T^{2} + 22639 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 + 142100 T + 61463939406 T^{2} - 39567358829040000 T^{3} - \)\(14\!\cdots\!73\)\( T^{4} - 39567358829040000 p^{9} T^{5} + 61463939406 p^{18} T^{6} + 142100 p^{27} T^{7} + p^{36} T^{8} \)
19$D_4\times C_2$ \( 1 + 525481 T - 284309947637 T^{2} - 44631816259115360 T^{3} + \)\(12\!\cdots\!44\)\( T^{4} - 44631816259115360 p^{9} T^{5} - 284309947637 p^{18} T^{6} + 525481 p^{27} T^{7} + p^{36} T^{8} \)
23$D_4\times C_2$ \( 1 - 1793596 T - 340665497790 T^{2} + 80089825874056320 T^{3} + \)\(36\!\cdots\!59\)\( T^{4} + 80089825874056320 p^{9} T^{5} - 340665497790 p^{18} T^{6} - 1793596 p^{27} T^{7} + p^{36} T^{8} \)
29$D_{4}$ \( ( 1 - 424441 T + 28385852443318 T^{2} - 424441 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 - 7172866 T - 9736070318071 T^{2} - 59583797487788313210 T^{3} + \)\(16\!\cdots\!12\)\( T^{4} - 59583797487788313210 p^{9} T^{5} - 9736070318071 p^{18} T^{6} - 7172866 p^{27} T^{7} + p^{36} T^{8} \)
37$D_4\times C_2$ \( 1 + 9253927 T - 78846469846757 T^{2} - \)\(88\!\cdots\!36\)\( T^{3} - \)\(96\!\cdots\!66\)\( T^{4} - \)\(88\!\cdots\!36\)\( p^{9} T^{5} - 78846469846757 p^{18} T^{6} + 9253927 p^{27} T^{7} + p^{36} T^{8} \)
41$D_{4}$ \( ( 1 - 26900706 T + 798410053286242 T^{2} - 26900706 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
43$D_{4}$ \( ( 1 + 17273717 T + 466964586678636 T^{2} + 17273717 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 - 629472 T - 233891249742682 T^{2} + \)\(12\!\cdots\!96\)\( T^{3} - \)\(11\!\cdots\!61\)\( T^{4} + \)\(12\!\cdots\!96\)\( p^{9} T^{5} - 233891249742682 p^{18} T^{6} - 629472 p^{27} T^{7} + p^{36} T^{8} \)
53$D_4\times C_2$ \( 1 - 148593939 T + 10081378867430531 T^{2} - \)\(80\!\cdots\!36\)\( T^{3} + \)\(60\!\cdots\!18\)\( T^{4} - \)\(80\!\cdots\!36\)\( p^{9} T^{5} + 10081378867430531 p^{18} T^{6} - 148593939 p^{27} T^{7} + p^{36} T^{8} \)
59$D_4\times C_2$ \( 1 - 235133615 T + 27265537081584711 T^{2} - \)\(25\!\cdots\!40\)\( T^{3} + \)\(23\!\cdots\!00\)\( T^{4} - \)\(25\!\cdots\!40\)\( p^{9} T^{5} + 27265537081584711 p^{18} T^{6} - 235133615 p^{27} T^{7} + p^{36} T^{8} \)
61$D_4\times C_2$ \( 1 - 20565172 T - 18639632996191678 T^{2} + \)\(88\!\cdots\!40\)\( T^{3} + \)\(22\!\cdots\!19\)\( T^{4} + \)\(88\!\cdots\!40\)\( p^{9} T^{5} - 18639632996191678 p^{18} T^{6} - 20565172 p^{27} T^{7} + p^{36} T^{8} \)
67$D_4\times C_2$ \( 1 + 302146653 T + 16277551529424965 T^{2} + \)\(62\!\cdots\!50\)\( T^{3} + \)\(24\!\cdots\!04\)\( T^{4} + \)\(62\!\cdots\!50\)\( p^{9} T^{5} + 16277551529424965 p^{18} T^{6} + 302146653 p^{27} T^{7} + p^{36} T^{8} \)
71$D_{4}$ \( ( 1 + 223330648 T + 38220162879166138 T^{2} + 223330648 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 + 476323479 T + 52578026568382061 T^{2} + \)\(26\!\cdots\!66\)\( T^{3} + \)\(13\!\cdots\!98\)\( T^{4} + \)\(26\!\cdots\!66\)\( p^{9} T^{5} + 52578026568382061 p^{18} T^{6} + 476323479 p^{27} T^{7} + p^{36} T^{8} \)
79$D_4\times C_2$ \( 1 - 34422028 T - 122895108472547401 T^{2} + \)\(39\!\cdots\!84\)\( T^{3} + \)\(88\!\cdots\!68\)\( T^{4} + \)\(39\!\cdots\!84\)\( p^{9} T^{5} - 122895108472547401 p^{18} T^{6} - 34422028 p^{27} T^{7} + p^{36} T^{8} \)
83$D_{4}$ \( ( 1 + 277779421 T + 321002486728181176 T^{2} + 277779421 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 - 868201722 T - 99711490293281230 T^{2} - \)\(13\!\cdots\!12\)\( T^{3} + \)\(34\!\cdots\!59\)\( T^{4} - \)\(13\!\cdots\!12\)\( p^{9} T^{5} - 99711490293281230 p^{18} T^{6} - 868201722 p^{27} T^{7} + p^{36} T^{8} \)
97$D_{4}$ \( ( 1 + 2474830143 T + 3030317362453449256 T^{2} + 2474830143 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.224568510664180432912912744148, −7.82392905311917603551380939700, −7.49267824256504619542226582773, −6.96974260047948007964601119680, −6.92589353686988046488133072401, −6.75803804607938666193541099696, −6.67095214908729378204614046651, −6.14836069418806470004454376919, −5.66002906918767552710103056580, −5.35207320498047844866536145236, −5.35177748012155161038621756052, −4.46364304586880893617517637935, −4.27547207831455504207600059591, −4.15820229400793567168639385825, −3.94180118004675023645995594980, −3.20728842076904667869776254982, −2.95741501410315377983756710747, −2.66389834238602824084335791539, −2.66248602662865789708996309513, −2.00643473333992188935409415789, −1.28301205125753134657253330025, −1.12658712636416022263177304757, −0.927583274740358658384654118576, −0.32774158501000052054460177932, −0.06501301886286545921980180077, 0.06501301886286545921980180077, 0.32774158501000052054460177932, 0.927583274740358658384654118576, 1.12658712636416022263177304757, 1.28301205125753134657253330025, 2.00643473333992188935409415789, 2.66248602662865789708996309513, 2.66389834238602824084335791539, 2.95741501410315377983756710747, 3.20728842076904667869776254982, 3.94180118004675023645995594980, 4.15820229400793567168639385825, 4.27547207831455504207600059591, 4.46364304586880893617517637935, 5.35177748012155161038621756052, 5.35207320498047844866536145236, 5.66002906918767552710103056580, 6.14836069418806470004454376919, 6.67095214908729378204614046651, 6.75803804607938666193541099696, 6.92589353686988046488133072401, 6.96974260047948007964601119680, 7.49267824256504619542226582773, 7.82392905311917603551380939700, 8.224568510664180432912912744148

Graph of the $Z$-function along the critical line