Properties

Label 2-792-88.43-c1-0-46
Degree $2$
Conductor $792$
Sign $0.996 - 0.0820i$
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  + (1.37 + 0.331i)2-s + (1.78 + 0.910i)4-s − 2i·5-s + 3.02·7-s + (2.14 + 1.84i)8-s + (0.662 − 2.74i)10-s + (−2.33 + 2.35i)11-s + 1.32·13-s + (4.15 + i)14-s + (2.34 + 3.24i)16-s − 1.69i·17-s − 7.04i·19-s + (1.82 − 3.56i)20-s + (−3.98 + 2.47i)22-s + 3.12i·23-s + ⋯
L(s)  = 1  + (0.972 + 0.234i)2-s + (0.890 + 0.455i)4-s − 0.894i·5-s + 1.14·7-s + (0.759 + 0.650i)8-s + (0.209 − 0.869i)10-s + (−0.703 + 0.711i)11-s + 0.367·13-s + (1.10 + 0.267i)14-s + (0.585 + 0.810i)16-s − 0.411i·17-s − 1.61i·19-s + (0.407 − 0.796i)20-s + (−0.850 + 0.526i)22-s + 0.651i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0820i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0820i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(792\)    =    \(2^{3} \cdot 3^{2} \cdot 11\)
Sign: $0.996 - 0.0820i$
Analytic conductor: \(6.32415\)
Root analytic conductor: \(2.51478\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{792} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 792,\ (\ :1/2),\ 0.996 - 0.0820i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.14169 + 0.129031i\)
\(L(\frac12)\) \(\approx\) \(3.14169 + 0.129031i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.37 - 0.331i)T \)
3 \( 1 \)
11 \( 1 + (2.33 - 2.35i)T \)
good5 \( 1 + 2iT - 5T^{2} \)
7 \( 1 - 3.02T + 7T^{2} \)
13 \( 1 - 1.32T + 13T^{2} \)
17 \( 1 + 1.69iT - 17T^{2} \)
19 \( 1 + 7.04iT - 19T^{2} \)
23 \( 1 - 3.12iT - 23T^{2} \)
29 \( 1 + 1.54T + 29T^{2} \)
31 \( 1 - 8.30iT - 31T^{2} \)
37 \( 1 + 4.66iT - 37T^{2} \)
41 \( 1 + 7.73iT - 41T^{2} \)
43 \( 1 - 3.95iT - 43T^{2} \)
47 \( 1 + 6iT - 47T^{2} \)
53 \( 1 - 8.24iT - 53T^{2} \)
59 \( 1 + 11.9T + 59T^{2} \)
61 \( 1 + 8.10T + 61T^{2} \)
67 \( 1 - 6.24T + 67T^{2} \)
71 \( 1 - 12.2iT - 71T^{2} \)
73 \( 1 - 3.08iT - 73T^{2} \)
79 \( 1 - 12.4T + 79T^{2} \)
83 \( 1 + 4.71iT - 83T^{2} \)
89 \( 1 + 16.6T + 89T^{2} \)
97 \( 1 + 13.3T + 97T^{2} \)
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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

−10.68071398710248065479308936809, −9.253126612546877631552566178494, −8.453137990958174892566968660589, −7.57405446695000088378755394455, −6.87096843545751126740453408203, −5.39765890051931408357030446313, −5.00688256695540489659872578706, −4.22889891544584674515297565017, −2.75725373766453698303064917805, −1.51767190767838413111494390601, 1.62443288371988846463433342610, 2.80378986325135388681859307850, 3.79035643900365941917606926398, 4.83339372262381761708286184028, 5.84866369597369114585589786579, 6.47274699331840018462593006905, 7.77755782374693201615312479876, 8.158752157140861115945875236137, 9.765921307738481915548347036755, 10.77821171167380559894466993454

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