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Properties

Label 2-792-11.3-c1-0-14
Degree $2$
Conductor $792$
Sign $-0.859 + 0.511i$
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  + (−1.30 − 0.951i)5-s + (0.690 − 2.12i)7-s + (−2.80 + 1.76i)11-s + (0.190 − 0.138i)13-s + (−2.11 − 1.53i)17-s + (−1.11 − 3.44i)19-s − 3.47·23-s + (−0.736 − 2.26i)25-s + (−0.618 + 1.90i)29-s + (−2.5 + 1.81i)31-s + (−2.92 + 2.12i)35-s + (−2.07 + 6.37i)37-s + (−2.54 − 7.83i)41-s − 11.9·43-s + (−2.28 − 7.02i)47-s + ⋯
L(s)  = 1  + (−0.585 − 0.425i)5-s + (0.261 − 0.803i)7-s + (−0.846 + 0.531i)11-s + (0.0529 − 0.0384i)13-s + (−0.513 − 0.373i)17-s + (−0.256 − 0.789i)19-s − 0.723·23-s + (−0.147 − 0.453i)25-s + (−0.114 + 0.353i)29-s + (−0.449 + 0.326i)31-s + (−0.494 + 0.359i)35-s + (−0.340 + 1.04i)37-s + (−0.397 − 1.22i)41-s − 1.82·43-s + (−0.332 − 1.02i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.859 + 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(792\)    =    \(2^{3} \cdot 3^{2} \cdot 11\)
Sign: $-0.859 + 0.511i$
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: \((2,\ 792,\ (\ :1/2),\ -0.859 + 0.511i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.159828 - 0.581119i\)
\(L(\frac12)\) \(\approx\) \(0.159828 - 0.581119i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.80 - 1.76i)T \)
good5 \( 1 + (1.30 + 0.951i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-0.690 + 2.12i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-0.190 + 0.138i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (2.11 + 1.53i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.11 + 3.44i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 3.47T + 23T^{2} \)
29 \( 1 + (0.618 - 1.90i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (2.5 - 1.81i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.07 - 6.37i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.54 + 7.83i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 11.9T + 43T^{2} \)
47 \( 1 + (2.28 + 7.02i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-9.35 + 6.79i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.881 + 2.71i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-3.54 - 2.57i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 14.0T + 67T^{2} \)
71 \( 1 + (7.16 + 5.20i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-4.09 + 12.5i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-1.42 + 1.03i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-7.42 - 5.39i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 12.2T + 89T^{2} \)
97 \( 1 + (-4.78 + 3.47i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.15784623079596204636696789185, −8.939775474788635759915398831908, −8.199468943340440899585559440236, −7.37925453286861261923370057906, −6.64785456539416826654051675716, −5.18437285253534552712841686442, −4.54962889156596611235896591143, −3.51161543997683599302110364672, −2.04085543615239845620855467878, −0.28433181114560871987045309165, 1.98835962467985474878606126129, 3.16024367695056069917843379990, 4.20068911884552153463521786469, 5.44407149152177469809396418874, 6.13756889723816922420899491869, 7.32851293905154989878643184242, 8.130416795338834778083571356214, 8.739876558048406194654785589471, 9.870720511767403956030116766523, 10.71110216388648147263316641393

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