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Properties

Label 2-792-792.547-c1-0-58
Degree $2$
Conductor $792$
Sign $0.977 + 0.212i$
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  + (−1.40 + 0.147i)2-s + (−0.381 + 1.68i)3-s + (1.95 − 0.415i)4-s + (0.287 − 2.43i)6-s + (−2.68 + 0.874i)8-s + (−2.70 − 1.29i)9-s + (−2.27 − 2.41i)11-s + (−0.0444 + 3.46i)12-s + (3.65 − 1.62i)16-s + (2.51 − 3.46i)17-s + (4 + 1.41i)18-s + (8.28 − 2.69i)19-s + (3.55 + 3.06i)22-s + (−0.449 − 4.87i)24-s + (−4.89 − 1.03i)25-s + ⋯
L(s)  = 1  + (−0.994 + 0.104i)2-s + (−0.220 + 0.975i)3-s + (0.978 − 0.207i)4-s + (0.117 − 0.993i)6-s + (−0.951 + 0.309i)8-s + (−0.902 − 0.430i)9-s + (−0.685 − 0.728i)11-s + (−0.0128 + 0.999i)12-s + (0.913 − 0.406i)16-s + (0.610 − 0.840i)17-s + (0.942 + 0.333i)18-s + (1.90 − 0.617i)19-s + (0.757 + 0.652i)22-s + (−0.0917 − 0.995i)24-s + (−0.978 − 0.207i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.212i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.977 + 0.212i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(792\)    =    \(2^{3} \cdot 3^{2} \cdot 11\)
Sign: $0.977 + 0.212i$
Analytic conductor: \(6.32415\)
Root analytic conductor: \(2.51478\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{792} (547, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 792,\ (\ :1/2),\ 0.977 + 0.212i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.759706 - 0.0815099i\)
\(L(\frac12)\) \(\approx\) \(0.759706 - 0.0815099i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 - 0.147i)T \)
3 \( 1 + (0.381 - 1.68i)T \)
11 \( 1 + (2.27 + 2.41i)T \)
good5 \( 1 + (4.89 + 1.03i)T^{2} \)
7 \( 1 + (-0.731 + 6.96i)T^{2} \)
13 \( 1 + (8.69 + 9.66i)T^{2} \)
17 \( 1 + (-2.51 + 3.46i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (-8.28 + 2.69i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.03 + 28.8i)T^{2} \)
31 \( 1 + (-20.7 - 23.0i)T^{2} \)
37 \( 1 + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (5.86 + 5.28i)T + (4.28 + 40.7i)T^{2} \)
43 \( 1 + (-7.84 - 4.52i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-42.9 - 19.1i)T^{2} \)
53 \( 1 + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-14.2 + 3.01i)T + (53.8 - 23.9i)T^{2} \)
61 \( 1 + (40.8 - 45.3i)T^{2} \)
67 \( 1 + (1.53 + 2.65i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (-15.5 - 5.04i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-77.2 + 16.4i)T^{2} \)
83 \( 1 + (-5.23 - 11.7i)T + (-55.5 + 61.6i)T^{2} \)
89 \( 1 + 17.8T + 89T^{2} \)
97 \( 1 + (1.65 + 15.7i)T + (-94.8 + 20.1i)T^{2} \)
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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

−9.925166477347011938113656467630, −9.659325655583562137747904035766, −8.676098063779385978515198205251, −7.86858638021821992711329380333, −6.95387197928510324022605602324, −5.67038994151906047813562228762, −5.22052848018945587544230166709, −3.55421491064154939398038250886, −2.66420826883961921772352775894, −0.62502423251129549512185080206, 1.17298009929241482132312230799, 2.26097673088986791964341177378, 3.45834706120501234226277907176, 5.35233294038928848164890864571, 6.08009175960483589935943344327, 7.24245939759197235167421223565, 7.67965072267564696266818890493, 8.396756226966981977916476478611, 9.558373409149989371304997386560, 10.19032415172313926377821378007

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