Properties

Label 2-28e2-1.1-c1-0-5
Degree $2$
Conductor $784$
Sign $1$
Analytic cond. $6.26027$
Root an. cond. $2.50205$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 3·5-s − 2·9-s + 3·11-s + 2·13-s − 3·15-s + 3·17-s + 19-s − 3·23-s + 4·25-s + 5·27-s − 6·29-s + 7·31-s − 3·33-s − 37-s − 2·39-s + 6·41-s + 4·43-s − 6·45-s + 9·47-s − 3·51-s + 3·53-s + 9·55-s − 57-s − 9·59-s − 61-s + 6·65-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.34·5-s − 2/3·9-s + 0.904·11-s + 0.554·13-s − 0.774·15-s + 0.727·17-s + 0.229·19-s − 0.625·23-s + 4/5·25-s + 0.962·27-s − 1.11·29-s + 1.25·31-s − 0.522·33-s − 0.164·37-s − 0.320·39-s + 0.937·41-s + 0.609·43-s − 0.894·45-s + 1.31·47-s − 0.420·51-s + 0.412·53-s + 1.21·55-s − 0.132·57-s − 1.17·59-s − 0.128·61-s + 0.744·65-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{784} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.648272273\)
\(L(\frac12)\) \(\approx\) \(1.648272273\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + T + p T^{2} \)
79 \( 1 - 13 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−10.28821554628842933399761045452, −9.465538249257530723796427509649, −8.832835367079797434619327214039, −7.68892345677099146953256519867, −6.40175986647281313923469473601, −5.96957053958209996558586270063, −5.22421181432809928276365314617, −3.86935901775791325818436521316, −2.51980819633185345991452121168, −1.19485794360651508706892872923, 1.19485794360651508706892872923, 2.51980819633185345991452121168, 3.86935901775791325818436521316, 5.22421181432809928276365314617, 5.96957053958209996558586270063, 6.40175986647281313923469473601, 7.68892345677099146953256519867, 8.832835367079797434619327214039, 9.465538249257530723796427509649, 10.28821554628842933399761045452

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