Properties

Degree $2$
Conductor $3136$
Sign $0.755 + 0.654i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3136\)    =    \(2^{6} \cdot 7^{2}\)
Sign: $0.755 + 0.654i$
Motivic weight: \(1\)
Character: $\chi_{3136} (3135, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3136,\ (\ :1/2),\ 0.755 + 0.654i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.154980669\)
\(L(\frac12)\) \(\approx\) \(1.154980669\)
\(L(\frac{3}{2})\) 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

−8.536428923651488807045918370325, −8.136834600189938614514045555944, −6.98781947117146771214434162195, −6.06830755076197169956216885626, −5.72989050044370266438141976182, −4.77816580377407432374688501531, −4.03709187508633558527002261586, −2.98877441346360831430960445748, −1.76310070209963987818281819017, −0.58600254898219169023906617950, 0.74451300334969369936724884526, 2.55818084243568342478986689104, 2.78938344211452363029074565718, 4.36755663456722172677625188865, 4.77543365454055639206429654275, 5.92695331126394683189090261644, 6.57260244251443030947066863262, 6.96433645252916185442211899555, 8.128745841724463020465797181739, 8.641668440583450616374179560073

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