Properties

Label 2-792-88.75-c0-0-0
Degree $2$
Conductor $792$
Sign $0.550 - 0.835i$
Analytic cond. $0.395259$
Root an. cond. $0.628696$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.309 + 0.951i)2-s + (−0.809 − 0.587i)4-s + (0.809 − 0.587i)8-s + (0.809 − 0.587i)11-s + (0.309 + 0.951i)16-s + (0.5 + 1.53i)17-s + (1.30 − 0.951i)19-s + (0.309 + 0.951i)22-s + (−0.809 + 0.587i)25-s − 32-s − 1.61·34-s + (0.499 + 1.53i)38-s + (0.5 − 0.363i)41-s + 0.618·43-s − 0.999·44-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)2-s + (−0.809 − 0.587i)4-s + (0.809 − 0.587i)8-s + (0.809 − 0.587i)11-s + (0.309 + 0.951i)16-s + (0.5 + 1.53i)17-s + (1.30 − 0.951i)19-s + (0.309 + 0.951i)22-s + (−0.809 + 0.587i)25-s − 32-s − 1.61·34-s + (0.499 + 1.53i)38-s + (0.5 − 0.363i)41-s + 0.618·43-s − 0.999·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.550 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.550 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(792\)    =    \(2^{3} \cdot 3^{2} \cdot 11\)
Sign: $0.550 - 0.835i$
Analytic conductor: \(0.395259\)
Root analytic conductor: \(0.628696\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{792} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 792,\ (\ :0),\ 0.550 - 0.835i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8273706295\)
\(L(\frac12)\) \(\approx\) \(0.8273706295\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 - 0.951i)T \)
3 \( 1 \)
11 \( 1 + (-0.809 + 0.587i)T \)
good5 \( 1 + (0.809 - 0.587i)T^{2} \)
7 \( 1 + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (0.809 + 0.587i)T^{2} \)
17 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
19 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.309 - 0.951i)T^{2} \)
41 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
43 \( 1 - 0.618T + T^{2} \)
47 \( 1 + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (0.809 + 0.587i)T^{2} \)
59 \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \)
61 \( 1 + (0.809 - 0.587i)T^{2} \)
67 \( 1 + 1.61T + T^{2} \)
71 \( 1 + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
79 \( 1 + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \)
89 \( 1 + 0.618T + T^{2} \)
97 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
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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

−10.46756545423740723141866820341, −9.468747824644091638777667486901, −8.943948421324790125508300250963, −7.954801449549138117505299441707, −7.26701973604653971690362006205, −6.19026843786343710257042285668, −5.62774743081545042905803644765, −4.41564810886384052064988578427, −3.40497491778558501238876564787, −1.36416317590207705578583067242, 1.31857430434776127440643131250, 2.69174794412313188263532661467, 3.75162344067135490516729953060, 4.72520165683107944567022386793, 5.78191450161745878799906852727, 7.22384830096375345010998812755, 7.81564347402769255457210750099, 8.981819456720351159032683222625, 9.650349743144352185016882181236, 10.17016646836883994171040440533

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