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Properties

Label 2-792-11.3-c1-0-9
Degree $2$
Conductor $792$
Sign $0.266 + 0.963i$
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  + (−1.95 − 1.41i)5-s + (−1.46 + 4.49i)7-s + (−0.960 − 3.17i)11-s + (4.86 − 3.53i)13-s + (0.720 + 0.523i)17-s + (−1.02 − 3.15i)19-s − 0.204·23-s + (0.254 + 0.782i)25-s + (2.95 − 9.10i)29-s + (7.37 − 5.36i)31-s + (9.22 − 6.70i)35-s + (0.445 − 1.37i)37-s + (−1.09 − 3.36i)41-s − 7.79·43-s + (2.83 + 8.72i)47-s + ⋯
L(s)  = 1  + (−0.873 − 0.634i)5-s + (−0.552 + 1.69i)7-s + (−0.289 − 0.957i)11-s + (1.34 − 0.980i)13-s + (0.174 + 0.126i)17-s + (−0.234 − 0.723i)19-s − 0.0427·23-s + (0.0508 + 0.156i)25-s + (0.549 − 1.69i)29-s + (1.32 − 0.962i)31-s + (1.55 − 1.13i)35-s + (0.0732 − 0.225i)37-s + (−0.170 − 0.526i)41-s − 1.18·43-s + (0.413 + 1.27i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(792\)    =    \(2^{3} \cdot 3^{2} \cdot 11\)
Sign: $0.266 + 0.963i$
Analytic conductor: \(6.32415\)
Root analytic conductor: \(2.51478\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{792} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 792,\ (\ :1/2),\ 0.266 + 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.813647 - 0.618953i\)
\(L(\frac12)\) \(\approx\) \(0.813647 - 0.618953i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
11 \( 1 + (0.960 + 3.17i)T \)
good5 \( 1 + (1.95 + 1.41i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (1.46 - 4.49i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-4.86 + 3.53i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.720 - 0.523i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.02 + 3.15i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 0.204T + 23T^{2} \)
29 \( 1 + (-2.95 + 9.10i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-7.37 + 5.36i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.445 + 1.37i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.09 + 3.36i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 7.79T + 43T^{2} \)
47 \( 1 + (-2.83 - 8.72i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (3.76 - 2.73i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.305 + 0.940i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (5.20 + 3.78i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 6.17T + 67T^{2} \)
71 \( 1 + (-5.50 - 3.99i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-1.19 + 3.67i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-3.67 + 2.66i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (8.08 + 5.87i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 0.930T + 89T^{2} \)
97 \( 1 + (2.36 - 1.71i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.03045075779388621763011439692, −9.016823643263776420916094661634, −8.343329450808167528729774012164, −7.995023136349683637760393215609, −6.20769916757055710908336554521, −5.89533735197567598627626403498, −4.71979438624809357996458698041, −3.49056468488393253141918887407, −2.58658903039649252917368850369, −0.56606209877191434202279702643, 1.36099902362795370589536358279, 3.30501003212499668127192111809, 3.88910678415755818244753283952, 4.81167014686490345048527026332, 6.59786770518630156590654865728, 6.85211881466225518465122642857, 7.75151541832917042325234573529, 8.626943633150981940919054957158, 9.915019119580136433225533702280, 10.43930006481316199017647075618

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