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Properties

Label 2-792-11.4-c1-0-8
Degree $2$
Conductor $792$
Sign $0.889 + 0.456i$
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  + (1.44 − 1.04i)5-s + (0.109 + 0.335i)7-s + (−2.91 − 1.58i)11-s + (4.48 + 3.26i)13-s + (3.08 − 2.24i)17-s + (1.42 − 4.39i)19-s + 7.00·23-s + (−0.564 + 1.73i)25-s + (−0.654 − 2.01i)29-s + (−2.79 − 2.03i)31-s + (0.508 + 0.369i)35-s + (−0.537 − 1.65i)37-s + (0.458 − 1.41i)41-s − 3.92·43-s + (0.374 − 1.15i)47-s + ⋯
L(s)  = 1  + (0.644 − 0.468i)5-s + (0.0412 + 0.126i)7-s + (−0.878 − 0.477i)11-s + (1.24 + 0.904i)13-s + (0.748 − 0.543i)17-s + (0.327 − 1.00i)19-s + 1.46·23-s + (−0.112 + 0.347i)25-s + (−0.121 − 0.374i)29-s + (−0.501 − 0.364i)31-s + (0.0860 + 0.0625i)35-s + (−0.0884 − 0.272i)37-s + (0.0715 − 0.220i)41-s − 0.599·43-s + (0.0545 − 0.167i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.889 + 0.456i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.889 + 0.456i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(792\)    =    \(2^{3} \cdot 3^{2} \cdot 11\)
Sign: $0.889 + 0.456i$
Analytic conductor: \(6.32415\)
Root analytic conductor: \(2.51478\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{792} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 792,\ (\ :1/2),\ 0.889 + 0.456i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.73075 - 0.417961i\)
\(L(\frac12)\) \(\approx\) \(1.73075 - 0.417961i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.91 + 1.58i)T \)
good5 \( 1 + (-1.44 + 1.04i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (-0.109 - 0.335i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-4.48 - 3.26i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-3.08 + 2.24i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.42 + 4.39i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 7.00T + 23T^{2} \)
29 \( 1 + (0.654 + 2.01i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (2.79 + 2.03i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.537 + 1.65i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-0.458 + 1.41i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 3.92T + 43T^{2} \)
47 \( 1 + (-0.374 + 1.15i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-6.41 - 4.65i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.954 - 2.93i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-6.15 + 4.47i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 1.17T + 67T^{2} \)
71 \( 1 + (1.91 - 1.39i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-1.20 - 3.71i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (3.32 + 2.41i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-0.662 + 0.481i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 1.92T + 89T^{2} \)
97 \( 1 + (-7.66 - 5.57i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.19093371046930936793762731477, −9.135990530716543359630224451278, −8.826890422779686384599884144174, −7.63856412100624913400366620315, −6.71796173824691456303924177917, −5.61184742947426384584357648678, −5.07778487066202070611443894302, −3.70956891555598914775423130326, −2.51409589620288280832320284429, −1.08360072940221514239057928568, 1.37523813152350981912788051997, 2.80365923891170629527684248637, 3.73992224223106119432768818117, 5.23389367235893207188183958201, 5.83406584413856982095549996065, 6.85580730965703484865016938369, 7.82907372750415031115148710651, 8.546723777197764185495864190624, 9.690377825326522562185266908188, 10.47974278206140742120746610613

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