Properties

Label 8-55e4-1.1-c7e4-0-0
Degree $8$
Conductor $9150625$
Sign $1$
Analytic cond. $87138.8$
Root an. cond. $4.14501$
Motivic weight $7$
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  + 166·3-s + 1.37e4·9-s − 3.27e4·16-s − 1.82e5·23-s − 1.00e5·25-s + 7.80e5·27-s − 1.12e6·37-s + 2.62e6·47-s − 5.43e6·48-s + 7.26e5·53-s − 1.36e6·67-s − 3.03e7·69-s + 2.29e7·71-s − 1.67e7·75-s + 3.14e7·81-s + 3.03e7·97-s − 4.36e7·103-s − 1.86e8·111-s − 2.80e7·113-s + 3.89e7·121-s + 127-s + 131-s + 137-s + 139-s + 4.36e8·141-s − 4.51e8·144-s + 149-s + ⋯
L(s)  = 1  + 3.54·3-s + 6.29·9-s − 2·16-s − 3.13·23-s − 1.29·25-s + 7.63·27-s − 3.65·37-s + 3.69·47-s − 7.09·48-s + 0.670·53-s − 0.556·67-s − 11.1·69-s + 7.59·71-s − 4.57·75-s + 6.56·81-s + 3.37·97-s − 3.93·103-s − 12.9·111-s − 1.82·113-s + 2·121-s + 13.1·141-s − 12.5·144-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9150625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9150625 ^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(9150625\)    =    \(5^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(87138.8\)
Root analytic conductor: \(4.14501\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 9150625,\ (\ :7/2, 7/2, 7/2, 7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(2.831501184\)
\(L(\frac12)\) \(\approx\) \(2.831501184\)
\(L(\frac{9}{2})\) 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)$
bad5$C_2^2$ \( 1 + 100799 T^{2} + p^{14} T^{4} \)
11$C_2$ \( ( 1 - p^{7} T^{2} )^{2} \)
good2$C_2$ \( ( 1 - p^{4} T + p^{7} T^{2} )^{2}( 1 + p^{4} T + p^{7} T^{2} )^{2} \)
3$C_2$$\times$$C_2^2$ \( ( 1 - 83 T + p^{7} T^{2} )^{2}( 1 + 2515 T^{2} + p^{14} T^{4} ) \)
7$C_2^2$ \( ( 1 + p^{14} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + p^{14} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + p^{14} T^{4} )^{2} \)
19$C_2$ \( ( 1 + p^{7} T^{2} )^{4} \)
23$C_2$$\times$$C_2^2$ \( ( 1 + 91467 T + p^{7} T^{2} )^{2}( 1 + 1556561195 T^{2} + p^{14} T^{4} ) \)
29$C_2$ \( ( 1 + p^{7} T^{2} )^{4} \)
31$C_2^2$ \( ( 1 - 53499153997 T^{2} + p^{14} T^{4} )^{2} \)
37$C_2$$\times$$C_2^2$ \( ( 1 + 562471 T + p^{7} T^{2} )^{2}( 1 + 126509871575 T^{2} + p^{14} T^{4} ) \)
41$C_2$ \( ( 1 - p^{7} T^{2} )^{4} \)
43$C_2^2$ \( ( 1 + p^{14} T^{4} )^{2} \)
47$C_2$$\times$$C_2^2$ \( ( 1 - 1314924 T + p^{7} T^{2} )^{2}( 1 + 715778884850 T^{2} + p^{14} T^{4} ) \)
53$C_2$$\times$$C_2^2$ \( ( 1 - 363378 T + p^{7} T^{2} )^{2}( 1 - 2217378708790 T^{2} + p^{14} T^{4} ) \)
59$C_2$ \( ( 1 - 149955 T + p^{7} T^{2} )^{2}( 1 + 149955 T + p^{7} T^{2} )^{2} \)
61$C_2$ \( ( 1 - p^{7} T^{2} )^{4} \)
67$C_2$$\times$$C_2^2$ \( ( 1 + 684671 T + p^{7} T^{2} )^{2}( 1 - 11652648832405 T^{2} + p^{14} T^{4} ) \)
71$C_2$ \( ( 1 - 5729217 T + p^{7} T^{2} )^{4} \)
73$C_2^2$ \( ( 1 + p^{14} T^{4} )^{2} \)
79$C_2$ \( ( 1 + p^{7} T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + p^{14} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 68889823168417 T^{2} + p^{14} T^{4} )^{2} \)
97$C_2$$\times$$C_2^2$ \( ( 1 - 15182479 T + p^{7} T^{2} )^{2}( 1 + 68911099629215 T^{2} + p^{14} T^{4} ) \)
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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

−9.676548724405280285222075492547, −9.325487390309525345946482492259, −9.216002241984453895202842638851, −8.671652573542763446093535856830, −8.623305580720443978634534735115, −8.461851012149257660061091109304, −8.007911729221429605182872154662, −7.67134614615162769769498549956, −7.53411683606553157580810437323, −7.08219319128491771093702837229, −6.58929776283056764316388098673, −6.46815000265896759846777827090, −5.76036339020634153639211262259, −5.25575285130867725497750717489, −4.89314080904731790825733896516, −4.03055941455226570652858388577, −3.96273547626592944494209085612, −3.71640874137854895223041556128, −3.45053645405231032589920153225, −2.49228561313982853204118296028, −2.43650538720244338350380369073, −2.05356486478623011471357899241, −2.03720891111922466739087719674, −1.13643837574193612747517791388, −0.17982925035461033261190325651, 0.17982925035461033261190325651, 1.13643837574193612747517791388, 2.03720891111922466739087719674, 2.05356486478623011471357899241, 2.43650538720244338350380369073, 2.49228561313982853204118296028, 3.45053645405231032589920153225, 3.71640874137854895223041556128, 3.96273547626592944494209085612, 4.03055941455226570652858388577, 4.89314080904731790825733896516, 5.25575285130867725497750717489, 5.76036339020634153639211262259, 6.46815000265896759846777827090, 6.58929776283056764316388098673, 7.08219319128491771093702837229, 7.53411683606553157580810437323, 7.67134614615162769769498549956, 8.007911729221429605182872154662, 8.461851012149257660061091109304, 8.623305580720443978634534735115, 8.671652573542763446093535856830, 9.216002241984453895202842638851, 9.325487390309525345946482492259, 9.676548724405280285222075492547

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