Properties

Degree $8$
Conductor $1475789056$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·4-s + 6·5-s − 4·8-s + 3·9-s − 12·10-s + 8·16-s + 6·17-s − 6·18-s + 12·20-s + 11·25-s + 16·29-s − 8·32-s − 12·34-s + 6·36-s − 6·37-s − 24·40-s + 18·45-s − 22·50-s + 2·53-s − 32·58-s + 18·61-s + 8·64-s + 12·68-s − 12·72-s − 30·73-s + 12·74-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s + 2.68·5-s − 1.41·8-s + 9-s − 3.79·10-s + 2·16-s + 1.45·17-s − 1.41·18-s + 2.68·20-s + 11/5·25-s + 2.97·29-s − 1.41·32-s − 2.05·34-s + 36-s − 0.986·37-s − 3.79·40-s + 2.68·45-s − 3.11·50-s + 0.274·53-s − 4.20·58-s + 2.30·61-s + 64-s + 1.45·68-s − 1.41·72-s − 3.51·73-s + 1.39·74-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 7^{8}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{196} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.33175\)
\(L(\frac12)\) \(\approx\) \(1.33175\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
7 \( 1 \)
good3$C_2$$\times$$C_2^2$ \( ( 1 - p T^{2} )^{2}( 1 + p T^{2} + p^{2} T^{4} ) \)
5$C_2^2$ \( ( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \)
11$C_2^3$ \( 1 + 21 T^{2} + 320 T^{4} + 21 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2^2$ \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - 3 T + 20 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \)
19$C_2^2$$\times$$C_2^2$ \( ( 1 - 37 T^{2} + p^{2} T^{4} )( 1 + 26 T^{2} + p^{2} T^{4} ) \)
23$C_2^3$ \( 1 + 45 T^{2} + 1496 T^{4} + 45 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
31$C_2^2$$\times$$C_2^2$ \( ( 1 - 46 T^{2} + p^{2} T^{4} )( 1 - 13 T^{2} + p^{2} T^{4} ) \)
37$C_2^2$ \( ( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^3$ \( 1 - 19 T^{2} - 1848 T^{4} - 19 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2^2$ \( ( 1 - T - 52 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \)
59$C_2^3$ \( 1 - 91 T^{2} + 4800 T^{4} - 91 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2^2$ \( ( 1 - 9 T + 88 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^3$ \( 1 + 125 T^{2} + 11136 T^{4} + 125 p^{2} T^{6} + p^{4} T^{8} \)
71$C_2^2$ \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 + 15 T + 148 T^{2} + 15 p T^{3} + p^{2} T^{4} )^{2} \)
79$C_2^3$ \( 1 + 77 T^{2} - 312 T^{4} + 77 p^{2} T^{6} + p^{4} T^{8} \)
83$C_2^2$ \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + 27 T + 332 T^{2} + 27 p T^{3} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 + 106 T^{2} + p^{2} T^{4} )^{2} \)
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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

−9.187614351117016647491275895634, −9.041128515753374920510891953853, −8.749541074546534054826803460506, −8.428552086087957708997344353529, −8.191299020182837639562663996814, −7.84263370032142427596333648976, −7.77953917178740500088383962357, −7.12961729600125241607792861492, −7.01701416665119670608538140412, −6.49930992977061403608418558978, −6.45062265678780178289965525418, −6.38658450215870992379121208599, −5.65848175581371508819797104475, −5.50641057379856802006164589792, −5.42710271713889409029199171721, −5.26238408978096231976593731906, −4.38760915688068305876352430193, −4.24586697932127391967279326414, −3.79688584054174697644332884819, −3.03069037723054284426661923670, −2.77137164377651728596220986912, −2.64570715967625067272544901612, −1.78396757973552578333510768553, −1.54088698623920149509906363900, −1.06675956187295801218069986067, 1.06675956187295801218069986067, 1.54088698623920149509906363900, 1.78396757973552578333510768553, 2.64570715967625067272544901612, 2.77137164377651728596220986912, 3.03069037723054284426661923670, 3.79688584054174697644332884819, 4.24586697932127391967279326414, 4.38760915688068305876352430193, 5.26238408978096231976593731906, 5.42710271713889409029199171721, 5.50641057379856802006164589792, 5.65848175581371508819797104475, 6.38658450215870992379121208599, 6.45062265678780178289965525418, 6.49930992977061403608418558978, 7.01701416665119670608538140412, 7.12961729600125241607792861492, 7.77953917178740500088383962357, 7.84263370032142427596333648976, 8.191299020182837639562663996814, 8.428552086087957708997344353529, 8.749541074546534054826803460506, 9.041128515753374920510891953853, 9.187614351117016647491275895634

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