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Properties

Label 2-792-8.5-c1-0-5
Degree $2$
Conductor $792$
Sign $-0.944 + 0.328i$
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  + (−1.13 + 0.838i)2-s + (0.592 − 1.91i)4-s + 3.89i·5-s − 1.77·7-s + (0.928 + 2.67i)8-s + (−3.26 − 4.42i)10-s + i·11-s + 5.27i·13-s + (2.02 − 1.48i)14-s + (−3.29 − 2.26i)16-s + 2.76·17-s − 4.13i·19-s + (7.43 + 2.30i)20-s + (−0.838 − 1.13i)22-s − 6.91·23-s + ⋯
L(s)  = 1  + (−0.805 + 0.593i)2-s + (0.296 − 0.955i)4-s + 1.73i·5-s − 0.670·7-s + (0.328 + 0.944i)8-s + (−1.03 − 1.40i)10-s + 0.301i·11-s + 1.46i·13-s + (0.540 − 0.397i)14-s + (−0.824 − 0.565i)16-s + 0.669·17-s − 0.948i·19-s + (1.66 + 0.515i)20-s + (−0.178 − 0.242i)22-s − 1.44·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 + 0.328i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.944 + 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(792\)    =    \(2^{3} \cdot 3^{2} \cdot 11\)
Sign: $-0.944 + 0.328i$
Analytic conductor: \(6.32415\)
Root analytic conductor: \(2.51478\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{792} (397, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 792,\ (\ :1/2),\ -0.944 + 0.328i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0925359 - 0.548280i\)
\(L(\frac12)\) \(\approx\) \(0.0925359 - 0.548280i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.13 - 0.838i)T \)
3 \( 1 \)
11 \( 1 - iT \)
good5 \( 1 - 3.89iT - 5T^{2} \)
7 \( 1 + 1.77T + 7T^{2} \)
13 \( 1 - 5.27iT - 13T^{2} \)
17 \( 1 - 2.76T + 17T^{2} \)
19 \( 1 + 4.13iT - 19T^{2} \)
23 \( 1 + 6.91T + 23T^{2} \)
29 \( 1 + 1.74iT - 29T^{2} \)
31 \( 1 - 5.35T + 31T^{2} \)
37 \( 1 - 3.18iT - 37T^{2} \)
41 \( 1 - 2.49T + 41T^{2} \)
43 \( 1 + 2.49iT - 43T^{2} \)
47 \( 1 + 11.5T + 47T^{2} \)
53 \( 1 - 9.61iT - 53T^{2} \)
59 \( 1 - 9.27iT - 59T^{2} \)
61 \( 1 + 13.8iT - 61T^{2} \)
67 \( 1 + 3.84iT - 67T^{2} \)
71 \( 1 + 7.62T + 71T^{2} \)
73 \( 1 + 10.7T + 73T^{2} \)
79 \( 1 + 0.281T + 79T^{2} \)
83 \( 1 - 0.313iT - 83T^{2} \)
89 \( 1 - 2.47T + 89T^{2} \)
97 \( 1 - 11.7T + 97T^{2} \)
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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

−10.44096349661022465694960896330, −9.896376760365160249038422698839, −9.238835528706020103223507758348, −8.001840890135116749346519732508, −7.19821971137449764757934056893, −6.52715316392061401837170758339, −6.06384386680244491355630264395, −4.47061333858523118466655621000, −3.08681012882623575706944468024, −2.01747780794105059919817782176, 0.36184781938271914211976935897, 1.52774495506501313991056236272, 3.12196242715129764622960543054, 4.09827082532318359992862961302, 5.34211327191207280023292935683, 6.23097511241838619054266910125, 7.85838445681188085327774750634, 8.124421603854878096867526212303, 8.988976353385084257326978263742, 9.958009267048374897211618482149

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