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Properties

Label 2-792-24.11-c1-0-20
Degree $2$
Conductor $792$
Sign $-0.0872 - 0.996i$
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  + (1.08 + 0.910i)2-s + (0.341 + 1.97i)4-s + 2.67·5-s + 2.05i·7-s + (−1.42 + 2.44i)8-s + (2.89 + 2.43i)10-s − i·11-s − 0.0834i·13-s + (−1.87 + 2.22i)14-s + (−3.76 + 1.34i)16-s + 2.66i·17-s + 1.44·19-s + (0.912 + 5.26i)20-s + (0.910 − 1.08i)22-s − 2.02·23-s + ⋯
L(s)  = 1  + (0.765 + 0.643i)2-s + (0.170 + 0.985i)4-s + 1.19·5-s + 0.778i·7-s + (−0.503 + 0.863i)8-s + (0.914 + 0.769i)10-s − 0.301i·11-s − 0.0231i·13-s + (−0.501 + 0.595i)14-s + (−0.941 + 0.336i)16-s + 0.646i·17-s + 0.330·19-s + (0.203 + 1.17i)20-s + (0.194 − 0.230i)22-s − 0.422·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0872 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0872 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(792\)    =    \(2^{3} \cdot 3^{2} \cdot 11\)
Sign: $-0.0872 - 0.996i$
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: \((2,\ 792,\ (\ :1/2),\ -0.0872 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.86567 + 2.03620i\)
\(L(\frac12)\) \(\approx\) \(1.86567 + 2.03620i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.08 - 0.910i)T \)
3 \( 1 \)
11 \( 1 + iT \)
good5 \( 1 - 2.67T + 5T^{2} \)
7 \( 1 - 2.05iT - 7T^{2} \)
13 \( 1 + 0.0834iT - 13T^{2} \)
17 \( 1 - 2.66iT - 17T^{2} \)
19 \( 1 - 1.44T + 19T^{2} \)
23 \( 1 + 2.02T + 23T^{2} \)
29 \( 1 - 8.40T + 29T^{2} \)
31 \( 1 + 8.49iT - 31T^{2} \)
37 \( 1 + 3.12iT - 37T^{2} \)
41 \( 1 - 5.03iT - 41T^{2} \)
43 \( 1 + 12.0T + 43T^{2} \)
47 \( 1 - 1.86T + 47T^{2} \)
53 \( 1 - 5.28T + 53T^{2} \)
59 \( 1 - 10.1iT - 59T^{2} \)
61 \( 1 + 5.80iT - 61T^{2} \)
67 \( 1 + 2.55T + 67T^{2} \)
71 \( 1 + 0.947T + 71T^{2} \)
73 \( 1 - 7.14T + 73T^{2} \)
79 \( 1 + 6.37iT - 79T^{2} \)
83 \( 1 + 2.22iT - 83T^{2} \)
89 \( 1 + 11.3iT - 89T^{2} \)
97 \( 1 + 18.0T + 97T^{2} \)
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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

−10.43780356106407112438917148659, −9.553280638776264456749949736927, −8.673881599226927369350754454273, −7.941289838395771880481814856860, −6.69470657826529048262492281210, −5.96533807795878584547530086329, −5.45841861867774342045304486744, −4.33111205852245635543954488960, −3.00434848122055057468461634508, −2.02188448000900602990613455989, 1.19255129428878598155753393930, 2.35918160852014552198323226973, 3.47544978864858331337602946641, 4.69633922135267379963890959279, 5.38618610692442107533301122984, 6.46773565517070573733425602104, 7.08114625785998477274895411078, 8.559857027283828593326487703233, 9.663350380743487090913330887553, 10.12126242680631363695290667312

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