Properties

Label 2-936-104.77-c1-0-40
Degree $2$
Conductor $936$
Sign $0.384 - 0.923i$
Analytic cond. $7.47399$
Root an. cond. $2.73386$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.34 + 0.437i)2-s + (1.61 + 1.17i)4-s + 1.66·5-s + 3.57i·7-s + (1.66 + 2.28i)8-s + (2.23 + 0.726i)10-s + 1.02·11-s + (−1.87 − 3.07i)13-s + (−1.56 + 4.80i)14-s + (1.23 + 3.80i)16-s + 5.05·17-s − 1.16·19-s + (2.68 + 1.95i)20-s + (1.38 + 0.449i)22-s − 8.17·23-s + ⋯
L(s)  = 1  + (0.951 + 0.309i)2-s + (0.809 + 0.587i)4-s + 0.743·5-s + 1.35i·7-s + (0.587 + 0.809i)8-s + (0.707 + 0.229i)10-s + 0.309·11-s + (−0.520 − 0.853i)13-s + (−0.417 + 1.28i)14-s + (0.309 + 0.951i)16-s + 1.22·17-s − 0.266·19-s + (0.601 + 0.437i)20-s + (0.294 + 0.0957i)22-s − 1.70·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.384 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.384 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $0.384 - 0.923i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ 0.384 - 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.69277 + 1.79568i\)
\(L(\frac12)\) \(\approx\) \(2.69277 + 1.79568i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.34 - 0.437i)T \)
3 \( 1 \)
13 \( 1 + (1.87 + 3.07i)T \)
good5 \( 1 - 1.66T + 5T^{2} \)
7 \( 1 - 3.57iT - 7T^{2} \)
11 \( 1 - 1.02T + 11T^{2} \)
17 \( 1 - 5.05T + 17T^{2} \)
19 \( 1 + 1.16T + 19T^{2} \)
23 \( 1 + 8.17T + 23T^{2} \)
29 \( 1 - 4.29iT - 29T^{2} \)
31 \( 1 + 7.98iT - 31T^{2} \)
37 \( 1 - 9.83T + 37T^{2} \)
41 \( 1 - 1.62iT - 41T^{2} \)
43 \( 1 + 2.35iT - 43T^{2} \)
47 \( 1 - 7.73iT - 47T^{2} \)
53 \( 1 + 11.2iT - 53T^{2} \)
59 \( 1 - 9.73T + 59T^{2} \)
61 \( 1 + 11.4iT - 61T^{2} \)
67 \( 1 - 7.23T + 67T^{2} \)
71 \( 1 - 10.5iT - 71T^{2} \)
73 \( 1 + 4.41iT - 73T^{2} \)
79 \( 1 + 6.94T + 79T^{2} \)
83 \( 1 - 2.29T + 83T^{2} \)
89 \( 1 + 9.02iT - 89T^{2} \)
97 \( 1 + 11.5iT - 97T^{2} \)
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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

−10.07658657332850173485259318922, −9.542136055360956114508829950943, −8.262902671254988106911603778468, −7.74432132600833746998787415275, −6.39447610241180234997989755542, −5.75398032737662008315665847160, −5.29471008773076112038233487718, −3.98164927294549261678572572709, −2.78982996351309244262166005502, −1.98547914836171622009525293140, 1.24494447161561924757164829928, 2.37972371560328035167213193801, 3.78951878150467678515650876660, 4.35244830006680637660784180482, 5.53017424152506196396004677532, 6.32922016060331245974635352046, 7.14833222372990245013581069918, 7.961985262251116094918513146187, 9.533216816461733886970206725525, 10.04692609644976326271259446520

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