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Properties

Label 2-792-11.4-c1-0-3
Degree $2$
Conductor $792$
Sign $0.522 - 0.852i$
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  + (−0.722 + 0.524i)5-s + (0.697 + 2.14i)7-s + (1.80 − 2.78i)11-s + (2.00 + 1.45i)13-s + (−0.476 + 0.346i)17-s + (−0.793 + 2.44i)19-s − 0.671·23-s + (−1.29 + 3.99i)25-s + (1.08 + 3.32i)29-s + (5.65 + 4.10i)31-s + (−1.63 − 1.18i)35-s + (2.16 + 6.65i)37-s + (−2.84 + 8.76i)41-s + 2.14·43-s + (1.34 − 4.12i)47-s + ⋯
L(s)  = 1  + (−0.323 + 0.234i)5-s + (0.263 + 0.811i)7-s + (0.542 − 0.839i)11-s + (0.557 + 0.404i)13-s + (−0.115 + 0.0839i)17-s + (−0.182 + 0.560i)19-s − 0.139·23-s + (−0.259 + 0.799i)25-s + (0.200 + 0.617i)29-s + (1.01 + 0.738i)31-s + (−0.275 − 0.200i)35-s + (0.355 + 1.09i)37-s + (−0.444 + 1.36i)41-s + 0.326·43-s + (0.195 − 0.601i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(792\)    =    \(2^{3} \cdot 3^{2} \cdot 11\)
Sign: $0.522 - 0.852i$
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: \((2,\ 792,\ (\ :1/2),\ 0.522 - 0.852i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28351 + 0.718688i\)
\(L(\frac12)\) \(\approx\) \(1.28351 + 0.718688i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.80 + 2.78i)T \)
good5 \( 1 + (0.722 - 0.524i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (-0.697 - 2.14i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-2.00 - 1.45i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.476 - 0.346i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.793 - 2.44i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 0.671T + 23T^{2} \)
29 \( 1 + (-1.08 - 3.32i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-5.65 - 4.10i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.16 - 6.65i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (2.84 - 8.76i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 2.14T + 43T^{2} \)
47 \( 1 + (-1.34 + 4.12i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-5.78 - 4.20i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (4.24 + 13.0i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-3.63 + 2.64i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 3.49T + 67T^{2} \)
71 \( 1 + (-0.357 + 0.259i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-1.97 - 6.07i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (3.88 + 2.82i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-5.74 + 4.17i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 11.4T + 89T^{2} \)
97 \( 1 + (5.74 + 4.17i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.49495968912897900467508849474, −9.477963131544232562650343617696, −8.591593062553383902690168605559, −8.113689082101474417263288035784, −6.80594619796827172016606339757, −6.10610231685937046486747022145, −5.11307593711073504853923467035, −3.91091693910068504607149881306, −2.95171745938112540777227570319, −1.46697961361752277375789307472, 0.824951746860878851841306532387, 2.38258565443220397398810963417, 3.98116100753070920532105287632, 4.42952739060391525398579785839, 5.73840851078375773257342897930, 6.78474042022362608561269514601, 7.56789718457264049715794078822, 8.377385889509882146365772498146, 9.310070013414928046765482359741, 10.21002192654645720469809492908

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