Properties

Degree $2$
Conductor $784$
Sign $0.188 + 0.981i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 1.73i·5-s − 2·9-s − 1.73i·11-s − 1.73i·15-s − 5.19i·17-s + 7·19-s − 8.66i·23-s + 2.00·25-s − 5·27-s − 6·29-s + 5·31-s − 1.73i·33-s − 5·37-s + 6.92i·41-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.774i·5-s − 0.666·9-s − 0.522i·11-s − 0.447i·15-s − 1.26i·17-s + 1.60·19-s − 1.80i·23-s + 0.400·25-s − 0.962·27-s − 1.11·29-s + 0.898·31-s − 0.301i·33-s − 0.821·37-s + 1.08i·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.188 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.188 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.188 + 0.981i$
Motivic weight: \(1\)
Character: $\chi_{784} (783, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ 0.188 + 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25939 - 1.04013i\)
\(L(\frac12)\) \(\approx\) \(1.25939 - 1.04013i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 - T + 3T^{2} \)
5 \( 1 + 1.73iT - 5T^{2} \)
11 \( 1 + 1.73iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 5.19iT - 17T^{2} \)
19 \( 1 - 7T + 19T^{2} \)
23 \( 1 + 8.66iT - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 - 5T + 31T^{2} \)
37 \( 1 + 5T + 37T^{2} \)
41 \( 1 - 6.92iT - 41T^{2} \)
43 \( 1 + 3.46iT - 43T^{2} \)
47 \( 1 - 3T + 47T^{2} \)
53 \( 1 + 9T + 53T^{2} \)
59 \( 1 - 9T + 59T^{2} \)
61 \( 1 + 8.66iT - 61T^{2} \)
67 \( 1 - 5.19iT - 67T^{2} \)
71 \( 1 - 3.46iT - 71T^{2} \)
73 \( 1 - 1.73iT - 73T^{2} \)
79 \( 1 - 5.19iT - 79T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 - 12.1iT - 89T^{2} \)
97 \( 1 - 6.92iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.860002516453963791533339154698, −9.138308019286908159521128785271, −8.502887213813762192742809824189, −7.74749511872022700380690954945, −6.66902675232716157113402250767, −5.48587411695623371841732715980, −4.79363137965197666229912936064, −3.42832494404980997039994527689, −2.53217417965741757745191656103, −0.78327429415185054960029689807, 1.78157633969838083959998193263, 3.06593996374930817045727118253, 3.72130528961797749442251659621, 5.24060217905519585047031327708, 6.07617530251102485833902398952, 7.25451356725258149891540215935, 7.78018782298229755560295735743, 8.845778233884623005822856928439, 9.594033853931277846300165412456, 10.41708011885085121909774219107

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