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Properties

Label 2-792-9.4-c1-0-0
Degree $2$
Conductor $792$
Sign $-0.410 - 0.911i$
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  + (−0.215 − 1.71i)3-s + (−0.748 + 1.29i)5-s + (−1.94 − 3.36i)7-s + (−2.90 + 0.739i)9-s + (−0.5 − 0.866i)11-s + (−3.15 + 5.47i)13-s + (2.38 + 1.00i)15-s + 4.19·17-s − 1.38·19-s + (−5.36 + 4.05i)21-s + (−2.06 + 3.57i)23-s + (1.38 + 2.39i)25-s + (1.89 + 4.83i)27-s + (−1.00 − 1.73i)29-s + (−1.71 + 2.97i)31-s + ⋯
L(s)  = 1  + (−0.124 − 0.992i)3-s + (−0.334 + 0.579i)5-s + (−0.733 − 1.27i)7-s + (−0.969 + 0.246i)9-s + (−0.150 − 0.261i)11-s + (−0.876 + 1.51i)13-s + (0.616 + 0.259i)15-s + 1.01·17-s − 0.318·19-s + (−1.17 + 0.885i)21-s + (−0.430 + 0.745i)23-s + (0.276 + 0.478i)25-s + (0.364 + 0.931i)27-s + (−0.186 − 0.322i)29-s + (−0.308 + 0.534i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.410 - 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.410 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(792\)    =    \(2^{3} \cdot 3^{2} \cdot 11\)
Sign: $-0.410 - 0.911i$
Analytic conductor: \(6.32415\)
Root analytic conductor: \(2.51478\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{792} (265, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 792,\ (\ :1/2),\ -0.410 - 0.911i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.103798 + 0.160655i\)
\(L(\frac12)\) \(\approx\) \(0.103798 + 0.160655i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.215 + 1.71i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (0.748 - 1.29i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (1.94 + 3.36i)T + (-3.5 + 6.06i)T^{2} \)
13 \( 1 + (3.15 - 5.47i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 4.19T + 17T^{2} \)
19 \( 1 + 1.38T + 19T^{2} \)
23 \( 1 + (2.06 - 3.57i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.00 + 1.73i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.71 - 2.97i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 10.0T + 37T^{2} \)
41 \( 1 + (0.256 - 0.444i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.32 - 4.02i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.19 + 10.7i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 6.07T + 53T^{2} \)
59 \( 1 + (3.10 - 5.38i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.96 + 3.39i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.17 - 7.23i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.65T + 71T^{2} \)
73 \( 1 + 3.97T + 73T^{2} \)
79 \( 1 + (-5.79 - 10.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.94 + 5.09i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 11.3T + 89T^{2} \)
97 \( 1 + (-2.33 - 4.04i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.56242493813724741614140812749, −9.835901954000933909492576421568, −8.770379783985830365747690964128, −7.57665342055869122084496276085, −7.12941435868272266702209920366, −6.55791663000270283772666412451, −5.36592814649996752243939247398, −3.96830696957474591337575683764, −3.06283467435387991352633694148, −1.60226807081949280556940204798, 0.093707183848205730859366220907, 2.58790798486773168116208831845, 3.42660813436985093910496294994, 4.74174497271490749251650817760, 5.43688914912543101315131731796, 6.15872437820369107947019677744, 7.66087764878665061143656446193, 8.516373955471389488667345240391, 9.185668206504155659609402959732, 10.06613707867715562176921913102

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