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Properties

Label 2-792-88.43-c1-0-47
Degree $2$
Conductor $792$
Sign $-0.979 + 0.202i$
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 + (−1.58 − 4.41i)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 + (−0.338 − 0.941i)22-s − 1.06i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 + 0.202i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.979 + 0.202i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(792\)    =    \(2^{3} \cdot 3^{2} \cdot 11\)
Sign: $-0.979 + 0.202i$
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: \((2,\ 792,\ (\ :1/2),\ -0.979 + 0.202i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0666891 - 0.651177i\)
\(L(\frac12)\) \(\approx\) \(0.0666891 - 0.651177i\)
\(L(\frac{3}{2})\) 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

−9.708758232244516794254409029372, −9.185208312994311233975104647393, −8.522081711533516802982122154160, −7.37642917735282081853799746934, −6.71676066503172070880715796781, −5.06419976555468779549317807869, −4.36947793138191347084963996881, −3.09764281491793082832232871989, −1.87797013250409026934592418431, −0.40512564365465397620767700593, 1.67667132149575047671009735170, 3.19781763988487174399186630851, 4.45529826220840321935330031681, 5.82843806233462428421455790943, 6.34388577686130888031429661458, 7.36312022554439330801690983110, 7.84427910321719446960875598417, 9.200214174459553813162840812371, 9.564339703586645104111105950612, 10.56875330786496217706219951234

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