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Properties

Label 2-792-792.403-c1-0-38
Degree $2$
Conductor $792$
Sign $0.556 + 0.831i$
Analytic cond. $6.32415$
Root an. cond. $2.51478$
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  + (−0.575 − 1.29i)2-s + (−0.684 − 1.59i)3-s + (−1.33 + 1.48i)4-s + (−1.66 + 1.79i)6-s + (2.68 + 0.874i)8-s + (−2.06 + 2.17i)9-s + (0.417 + 3.29i)11-s + (3.28 + 1.11i)12-s + (−0.418 − 3.97i)16-s + (−0.395 − 0.545i)17-s + (4 + 1.41i)18-s + (6.81 + 2.21i)19-s + (4.01 − 2.43i)22-s + (−0.449 − 4.87i)24-s + (3.34 + 3.71i)25-s + ⋯
L(s)  = 1  + (−0.406 − 0.913i)2-s + (−0.394 − 0.918i)3-s + (−0.669 + 0.743i)4-s + (−0.678 + 0.734i)6-s + (0.951 + 0.309i)8-s + (−0.687 + 0.725i)9-s + (0.126 + 0.992i)11-s + (0.947 + 0.321i)12-s + (−0.104 − 0.994i)16-s + (−0.0960 − 0.132i)17-s + (0.942 + 0.333i)18-s + (1.56 + 0.507i)19-s + (0.855 − 0.518i)22-s + (−0.0917 − 0.995i)24-s + (0.669 + 0.743i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.556 + 0.831i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.556 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(792\)    =    \(2^{3} \cdot 3^{2} \cdot 11\)
Sign: $0.556 + 0.831i$
Analytic conductor: \(6.32415\)
Root analytic conductor: \(2.51478\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{792} (403, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 792,\ (\ :1/2),\ 0.556 + 0.831i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.853856 - 0.456068i\)
\(L(\frac12)\) \(\approx\) \(0.853856 - 0.456068i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.575 + 1.29i)T \)
3 \( 1 + (0.684 + 1.59i)T \)
11 \( 1 + (-0.417 - 3.29i)T \)
good5 \( 1 + (-3.34 - 3.71i)T^{2} \)
7 \( 1 + (6.39 + 2.84i)T^{2} \)
13 \( 1 + (-12.7 - 2.70i)T^{2} \)
17 \( 1 + (0.395 + 0.545i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-6.81 - 2.21i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (26.4 + 11.7i)T^{2} \)
31 \( 1 + (30.3 + 6.44i)T^{2} \)
37 \( 1 + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.48 + 6.99i)T + (-37.4 + 16.6i)T^{2} \)
43 \( 1 + (-10.2 - 5.90i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.91 - 46.7i)T^{2} \)
53 \( 1 + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-5.89 + 6.54i)T + (-6.16 - 58.6i)T^{2} \)
61 \( 1 + (-59.6 + 12.6i)T^{2} \)
67 \( 1 + (-5.96 - 10.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.204 - 0.0664i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (52.8 - 58.7i)T^{2} \)
83 \( 1 + (8.24 - 0.866i)T + (81.1 - 17.2i)T^{2} \)
89 \( 1 + 11.2T + 89T^{2} \)
97 \( 1 + (-14.6 + 6.52i)T + (64.9 - 72.0i)T^{2} \)
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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

−10.16636307688792291220237069982, −9.450081819212734815140094056242, −8.488858294152557883821942792741, −7.51471433608222741424562502252, −7.04750399602827119370532333263, −5.61631949700343937062409159936, −4.70663787282178205220915238405, −3.34332836937959616471256074738, −2.17123180180126737620328300496, −1.07523018192297183914536463563, 0.77014011425652150320729463002, 3.13833534667017526960820947602, 4.31909413174513769823004999466, 5.24183534106320676313907700158, 5.96029169608209457260290287522, 6.83571551455365558876928090120, 7.948275759139231221995264409116, 8.820279099061221507284578518273, 9.428454122005763137363768751437, 10.25895478569562982326600493919

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