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Properties

Label 2-792-24.11-c1-0-5
Degree $2$
Conductor $792$
Sign $0.424 - 0.905i$
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  + (−0.906 − 1.08i)2-s + (−0.356 + 1.96i)4-s + 1.01·5-s + 2.06i·7-s + (2.45 − 1.39i)8-s + (−0.922 − 1.10i)10-s + i·11-s + 4.15i·13-s + (2.24 − 1.87i)14-s + (−3.74 − 1.40i)16-s + 0.276i·17-s − 4.36·19-s + (−0.362 + 2.00i)20-s + (1.08 − 0.906i)22-s − 2.68·23-s + ⋯
L(s)  = 1  + (−0.641 − 0.767i)2-s + (−0.178 + 0.984i)4-s + 0.455·5-s + 0.781i·7-s + (0.869 − 0.494i)8-s + (−0.291 − 0.349i)10-s + 0.301i·11-s + 1.15i·13-s + (0.599 − 0.501i)14-s + (−0.936 − 0.350i)16-s + 0.0669i·17-s − 1.00·19-s + (−0.0810 + 0.447i)20-s + (0.231 − 0.193i)22-s − 0.559·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.424 - 0.905i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.424 - 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(792\)    =    \(2^{3} \cdot 3^{2} \cdot 11\)
Sign: $0.424 - 0.905i$
Analytic conductor: \(6.32415\)
Root analytic conductor: \(2.51478\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{792} (683, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 792,\ (\ :1/2),\ 0.424 - 0.905i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.726719 + 0.461922i\)
\(L(\frac12)\) \(\approx\) \(0.726719 + 0.461922i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.906 + 1.08i)T \)
3 \( 1 \)
11 \( 1 - iT \)
good5 \( 1 - 1.01T + 5T^{2} \)
7 \( 1 - 2.06iT - 7T^{2} \)
13 \( 1 - 4.15iT - 13T^{2} \)
17 \( 1 - 0.276iT - 17T^{2} \)
19 \( 1 + 4.36T + 19T^{2} \)
23 \( 1 + 2.68T + 23T^{2} \)
29 \( 1 - 2.66T + 29T^{2} \)
31 \( 1 - 3.59iT - 31T^{2} \)
37 \( 1 - 5.90iT - 37T^{2} \)
41 \( 1 - 0.0318iT - 41T^{2} \)
43 \( 1 + 5.51T + 43T^{2} \)
47 \( 1 + 6.55T + 47T^{2} \)
53 \( 1 - 4.13T + 53T^{2} \)
59 \( 1 - 3.22iT - 59T^{2} \)
61 \( 1 - 8.96iT - 61T^{2} \)
67 \( 1 - 12.5T + 67T^{2} \)
71 \( 1 + 3.80T + 71T^{2} \)
73 \( 1 - 9.10T + 73T^{2} \)
79 \( 1 - 11.0iT - 79T^{2} \)
83 \( 1 - 11.9iT - 83T^{2} \)
89 \( 1 + 18.3iT - 89T^{2} \)
97 \( 1 - 16.5T + 97T^{2} \)
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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

−10.22826849972841987676199218853, −9.705096696349516236061871453494, −8.799461346943209668273574181965, −8.273513171249951194365844643500, −7.03001716090303110995736445155, −6.19956425669072786232914806243, −4.87184717233565733277223174113, −3.86791371588694268359152825869, −2.50040170550034991999409656951, −1.70495761150449415055478666803, 0.52298767776249816479197471814, 2.08997726057505039277928513698, 3.78718829611716720268266258853, 4.98863424617964267416838964029, 5.92725755652695518901181128876, 6.63356570226466889570389254081, 7.71721916878763040814771684699, 8.230620006533585093760461232370, 9.266764764908458190223154956374, 10.14274166764763326028390879132

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