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Properties

Label 2-2664-296.117-c0-0-0
Degree $2$
Conductor $2664$
Sign $0.763 - 0.646i$
Analytic cond. $1.32950$
Root an. cond. $1.15304$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 0.707i)2-s − 1.00i·4-s + (−1 + i)5-s − 7-s + (−0.707 − 0.707i)8-s + 1.41i·10-s + 11-s + (−0.707 + 0.707i)13-s + (−0.707 + 0.707i)14-s − 1.00·16-s + (0.707 + 0.707i)17-s + (−0.707 + 0.707i)19-s + (1.00 + 1.00i)20-s + (0.707 − 0.707i)22-s + (0.707 + 0.707i)23-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s − 1.00i·4-s + (−1 + i)5-s − 7-s + (−0.707 − 0.707i)8-s + 1.41i·10-s + 11-s + (−0.707 + 0.707i)13-s + (−0.707 + 0.707i)14-s − 1.00·16-s + (0.707 + 0.707i)17-s + (−0.707 + 0.707i)19-s + (1.00 + 1.00i)20-s + (0.707 − 0.707i)22-s + (0.707 + 0.707i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2664\)    =    \(2^{3} \cdot 3^{2} \cdot 37\)
Sign: $0.763 - 0.646i$
Analytic conductor: \(1.32950\)
Root analytic conductor: \(1.15304\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2664} (2485, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2664,\ (\ :0),\ 0.763 - 0.646i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.045919590\)
\(L(\frac12)\) \(\approx\) \(1.045919590\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 \)
37 \( 1 + (-0.707 - 0.707i)T \)
good5 \( 1 + (1 - i)T - iT^{2} \)
7 \( 1 + T + T^{2} \)
11 \( 1 - T + T^{2} \)
13 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
17 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
19 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
23 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
29 \( 1 + (-1 - i)T + iT^{2} \)
31 \( 1 - iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + 1.41T + T^{2} \)
53 \( 1 - iT - T^{2} \)
59 \( 1 + (-1 + i)T - iT^{2} \)
61 \( 1 + (1.41 + 1.41i)T + iT^{2} \)
67 \( 1 + 1.41T + T^{2} \)
71 \( 1 - 1.41T + T^{2} \)
73 \( 1 - iT - T^{2} \)
79 \( 1 + iT^{2} \)
83 \( 1 - iT - T^{2} \)
89 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
97 \( 1 + (-1 - i)T + iT^{2} \)
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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

−9.472559713564156631207106183397, −8.410998577678810418615313007699, −7.36535524830670398608775862261, −6.52733495964754336849084250535, −6.34361854443983488670577414989, −5.00742235302265401360787886456, −4.04723334864038131897076291148, −3.47193780973884609096448516902, −2.86383398815634168972815678434, −1.50882746121846903581356758917, 0.54988187467455456925364592165, 2.69673579205513706652654966853, 3.46609488298536598015911113412, 4.43459390172165722077128311553, 4.82184871844349074808866633567, 5.91041284891705055915272880482, 6.64816779659899490497414897329, 7.36890426413933048865381406351, 8.077263844721136873787702657043, 8.822113518880359493705864604439

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