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Properties

Label 8-75e4-1.1-c17e4-0-0
Degree $8$
Conductor $31640625$
Sign $1$
Analytic cond. $3.56579\times 10^{8}$
Root an. cond. $11.7224$
Motivic weight $17$
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  + 8.56e4·4-s − 8.60e7·9-s − 1.97e9·11-s + 1.74e10·16-s − 1.60e11·19-s + 9.40e11·29-s + 6.80e12·31-s − 7.37e12·36-s − 2.26e14·41-s − 1.69e14·44-s + 4.04e14·49-s − 3.45e15·59-s + 5.56e15·61-s + 3.82e15·64-s − 2.69e16·71-s − 1.37e16·76-s − 8.75e15·79-s + 5.55e15·81-s − 1.39e16·89-s + 1.70e17·99-s + 8.05e16·101-s − 1.01e18·109-s + 8.05e16·116-s + 4.20e17·121-s + 5.82e17·124-s + 127-s + 131-s + ⋯
L(s)  = 1  + 0.653·4-s − 2/3·9-s − 2.77·11-s + 1.01·16-s − 2.16·19-s + 0.349·29-s + 1.43·31-s − 0.435·36-s − 4.43·41-s − 1.81·44-s + 1.74·49-s − 3.06·59-s + 3.71·61-s + 1.69·64-s − 4.95·71-s − 1.41·76-s − 0.649·79-s + 1/3·81-s − 0.375·89-s + 1.85·99-s + 0.740·101-s − 4.88·109-s + 0.228·116-s + 0.832·121-s + 0.935·124-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+17/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(31640625\)    =    \(3^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(3.56579\times 10^{8}\)
Root analytic conductor: \(11.7224\)
Motivic weight: \(17\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 31640625,\ (\ :17/2, 17/2, 17/2, 17/2),\ 1)\)

Particular Values

\(L(9)\) \(\approx\) \(0.1197398879\)
\(L(\frac12)\) \(\approx\) \(0.1197398879\)
\(L(\frac{19}{2})\) 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)$
bad3$C_2$ \( ( 1 + p^{16} T^{2} )^{2} \)
5 \( 1 \)
good2$D_4\times C_2$ \( 1 - 21407 p^{2} T^{2} - 9835383 p^{10} T^{4} - 21407 p^{36} T^{6} + p^{68} T^{8} \)
7$D_4\times C_2$ \( 1 - 404791665498524 T^{2} + \)\(33\!\cdots\!38\)\( p^{4} T^{4} - 404791665498524 p^{34} T^{6} + p^{68} T^{8} \)
11$D_{4}$ \( ( 1 + 89777592 p T + 10351710384952534 p^{2} T^{2} + 89777592 p^{18} T^{3} + p^{34} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 28425949426053664172 T^{2} + \)\(20\!\cdots\!22\)\( p^{2} T^{4} - 28425949426053664172 p^{34} T^{6} + p^{68} T^{8} \)
17$D_4\times C_2$ \( 1 - \)\(26\!\cdots\!60\)\( T^{2} + \)\(29\!\cdots\!58\)\( T^{4} - \)\(26\!\cdots\!60\)\( p^{34} T^{6} + p^{68} T^{8} \)
19$D_{4}$ \( ( 1 + 80053542184 T + \)\(10\!\cdots\!58\)\( T^{2} + 80053542184 p^{17} T^{3} + p^{34} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - \)\(31\!\cdots\!56\)\( T^{2} + \)\(18\!\cdots\!18\)\( T^{4} - \)\(31\!\cdots\!56\)\( p^{34} T^{6} + p^{68} T^{8} \)
29$D_{4}$ \( ( 1 - 470374069572 T + \)\(14\!\cdots\!38\)\( T^{2} - 470374069572 p^{17} T^{3} + p^{34} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 - 3400754454592 T + \)\(45\!\cdots\!22\)\( T^{2} - 3400754454592 p^{17} T^{3} + p^{34} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - \)\(88\!\cdots\!44\)\( T^{2} + \)\(56\!\cdots\!18\)\( T^{4} - \)\(88\!\cdots\!44\)\( p^{34} T^{6} + p^{68} T^{8} \)
41$D_{4}$ \( ( 1 + 113376799448748 T + \)\(78\!\cdots\!02\)\( T^{2} + 113376799448748 p^{17} T^{3} + p^{34} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - \)\(12\!\cdots\!80\)\( T^{2} + \)\(90\!\cdots\!98\)\( T^{4} - \)\(12\!\cdots\!80\)\( p^{34} T^{6} + p^{68} T^{8} \)
47$D_4\times C_2$ \( 1 - \)\(66\!\cdots\!20\)\( T^{2} + \)\(24\!\cdots\!38\)\( T^{4} - \)\(66\!\cdots\!20\)\( p^{34} T^{6} + p^{68} T^{8} \)
53$D_4\times C_2$ \( 1 - \)\(64\!\cdots\!16\)\( T^{2} + \)\(18\!\cdots\!78\)\( T^{4} - \)\(64\!\cdots\!16\)\( p^{34} T^{6} + p^{68} T^{8} \)
59$D_{4}$ \( ( 1 + 1727524231086456 T + \)\(23\!\cdots\!58\)\( T^{2} + 1727524231086456 p^{17} T^{3} + p^{34} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 - 2784287656027900 T + \)\(64\!\cdots\!98\)\( T^{2} - 2784287656027900 p^{17} T^{3} + p^{34} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - \)\(57\!\cdots\!40\)\( p T^{2} + \)\(61\!\cdots\!58\)\( T^{4} - \)\(57\!\cdots\!40\)\( p^{35} T^{6} + p^{68} T^{8} \)
71$D_{4}$ \( ( 1 + 13489402206504816 T + \)\(10\!\cdots\!46\)\( T^{2} + 13489402206504816 p^{17} T^{3} + p^{34} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 + \)\(16\!\cdots\!64\)\( T^{2} + \)\(10\!\cdots\!98\)\( T^{4} + \)\(16\!\cdots\!64\)\( p^{34} T^{6} + p^{68} T^{8} \)
79$D_{4}$ \( ( 1 + 4376041565214880 T + \)\(25\!\cdots\!18\)\( T^{2} + 4376041565214880 p^{17} T^{3} + p^{34} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - \)\(82\!\cdots\!92\)\( T^{2} + \)\(47\!\cdots\!58\)\( T^{4} - \)\(82\!\cdots\!92\)\( p^{34} T^{6} + p^{68} T^{8} \)
89$D_{4}$ \( ( 1 + 6972184096107444 T + \)\(10\!\cdots\!98\)\( T^{2} + 6972184096107444 p^{17} T^{3} + p^{34} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - \)\(63\!\cdots\!40\)\( T^{2} + \)\(70\!\cdots\!38\)\( T^{4} - \)\(63\!\cdots\!40\)\( p^{34} T^{6} + p^{68} T^{8} \)
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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

−7.36598358669782451336423556656, −7.24872588741153052257146259112, −6.99859491345439352981814204849, −6.45912564168433579462045611864, −6.40515115920955940481325381899, −6.07788262425394028413110675221, −5.73757433226548859528569932830, −5.27082910065616474781515161038, −5.23323194240841289462164232107, −5.05163300447237961846522705937, −4.58157400579155726835990359301, −4.28422158606344547017192508413, −3.85549096330936112084398019100, −3.61433023402453684535402310627, −3.19849291994449255880117639074, −2.75316222318214366326603535075, −2.69609969212987938840225589975, −2.42467515805633998231852334885, −2.37160116115960980139615725150, −1.56040752652591603997548901871, −1.52440237776210627086701444196, −1.37286824065228699034837648216, −0.62900685106483919724106571850, −0.32882345626207830342872968180, −0.05873915200154127163503481477, 0.05873915200154127163503481477, 0.32882345626207830342872968180, 0.62900685106483919724106571850, 1.37286824065228699034837648216, 1.52440237776210627086701444196, 1.56040752652591603997548901871, 2.37160116115960980139615725150, 2.42467515805633998231852334885, 2.69609969212987938840225589975, 2.75316222318214366326603535075, 3.19849291994449255880117639074, 3.61433023402453684535402310627, 3.85549096330936112084398019100, 4.28422158606344547017192508413, 4.58157400579155726835990359301, 5.05163300447237961846522705937, 5.23323194240841289462164232107, 5.27082910065616474781515161038, 5.73757433226548859528569932830, 6.07788262425394028413110675221, 6.40515115920955940481325381899, 6.45912564168433579462045611864, 6.99859491345439352981814204849, 7.24872588741153052257146259112, 7.36598358669782451336423556656

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