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Properties

Label 2-792-88.43-c1-0-31
Degree $2$
Conductor $792$
Sign $0.586 - 0.809i$
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  + (0.927 + 1.06i)2-s + (−0.280 + 1.98i)4-s − 2i·5-s + 0.936·7-s + (−2.37 + 1.53i)8-s + (2.13 − 1.85i)10-s + (3.09 − 1.19i)11-s + 4.27·13-s + (0.868 + 0.999i)14-s + (−3.84 − 1.11i)16-s + 3.33i·17-s + 2.89i·19-s + (3.96 + 0.561i)20-s + (4.14 + 2.19i)22-s − 5.12i·23-s + ⋯
L(s)  = 1  + (0.655 + 0.755i)2-s + (−0.140 + 0.990i)4-s − 0.894i·5-s + 0.353·7-s + (−0.839 + 0.543i)8-s + (0.675 − 0.586i)10-s + (0.932 − 0.361i)11-s + 1.18·13-s + (0.232 + 0.267i)14-s + (−0.960 − 0.277i)16-s + 0.808i·17-s + 0.664i·19-s + (0.885 + 0.125i)20-s + (0.884 + 0.466i)22-s − 1.06i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.586 - 0.809i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.586 - 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(792\)    =    \(2^{3} \cdot 3^{2} \cdot 11\)
Sign: $0.586 - 0.809i$
Analytic conductor: \(6.32415\)
Root analytic conductor: \(2.51478\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{792} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 792,\ (\ :1/2),\ 0.586 - 0.809i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.10731 + 1.07591i\)
\(L(\frac12)\) \(\approx\) \(2.10731 + 1.07591i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.927 - 1.06i)T \)
3 \( 1 \)
11 \( 1 + (-3.09 + 1.19i)T \)
good5 \( 1 + 2iT - 5T^{2} \)
7 \( 1 - 0.936T + 7T^{2} \)
13 \( 1 - 4.27T + 13T^{2} \)
17 \( 1 - 3.33iT - 17T^{2} \)
19 \( 1 - 2.89iT - 19T^{2} \)
23 \( 1 + 5.12iT - 23T^{2} \)
29 \( 1 - 6.60T + 29T^{2} \)
31 \( 1 - 1.73iT - 31T^{2} \)
37 \( 1 - 6.18iT - 37T^{2} \)
41 \( 1 - 1.46iT - 41T^{2} \)
43 \( 1 - 10.3iT - 43T^{2} \)
47 \( 1 + 6iT - 47T^{2} \)
53 \( 1 + 8.24iT - 53T^{2} \)
59 \( 1 + 9.65T + 59T^{2} \)
61 \( 1 - 9.06T + 61T^{2} \)
67 \( 1 + 10.2T + 67T^{2} \)
71 \( 1 + 4.24iT - 71T^{2} \)
73 \( 1 + 13.2iT - 73T^{2} \)
79 \( 1 + 3.86T + 79T^{2} \)
83 \( 1 - 2.39iT - 83T^{2} \)
89 \( 1 + 3.47T + 89T^{2} \)
97 \( 1 - 11.3T + 97T^{2} \)
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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

−10.47232733335058882850365011168, −9.173286941820838002829916434846, −8.398089457370647549670765188491, −8.129943845609256195265444121043, −6.55555500175135224417902732559, −6.17390183299983517979849269040, −4.97312100741862327810036889909, −4.24773166382103875182546662892, −3.25807182953994622040438505305, −1.37310066846734640229155367655, 1.27788318315871964800276847796, 2.62687916222941356985628125369, 3.59735045193346706993218474089, 4.51532812623545281199652719350, 5.65551213860679786242790385505, 6.56004756049905768836581429190, 7.29263093655188580613641386331, 8.750860101579290565329673736462, 9.451222439991436409949420899447, 10.42137492151073899423323569560

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