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Properties

Label 2-504-504.403-c0-0-1
Degree $2$
Conductor $504$
Sign $0.971 + 0.235i$
Analytic cond. $0.251528$
Root an. cond. $0.501526$
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  + i·2-s − 3-s − 4-s + (−0.866 − 0.5i)5-s − i·6-s − i·7-s − i·8-s + 9-s + (0.5 − 0.866i)10-s + (0.5 + 0.866i)11-s + 12-s + (0.866 − 0.5i)13-s + 14-s + (0.866 + 0.5i)15-s + 16-s + (0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + i·2-s − 3-s − 4-s + (−0.866 − 0.5i)5-s − i·6-s − i·7-s − i·8-s + 9-s + (0.5 − 0.866i)10-s + (0.5 + 0.866i)11-s + 12-s + (0.866 − 0.5i)13-s + 14-s + (0.866 + 0.5i)15-s + 16-s + (0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.971 + 0.235i$
Analytic conductor: \(0.251528\)
Root analytic conductor: \(0.501526\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (403, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :0),\ 0.971 + 0.235i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4837417074\)
\(L(\frac12)\) \(\approx\) \(0.4837417074\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + T \)
7 \( 1 + iT \)
good5 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 - 2iT - T^{2} \)
83 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−11.11847323314820698440922546244, −10.15295785553606721950257362694, −9.306857326220757787907207699068, −8.058960658808242727174942574579, −7.38320011521472266620896251791, −6.61097744508091836758627332188, −5.56108583574107768908033895979, −4.37481272729121685199009028681, −4.05094465565953211715815421944, −0.76588111273126456913913258676, 1.66288235984031319277214187681, 3.48287038876852472229560576601, 4.13222332442139217877586155306, 5.65569622121145456965830989656, 6.18198229895547445248637464167, 7.75542194234591870291785121086, 8.620110418649082962331585751552, 9.609610684323911456379524875308, 10.67846571448001885872639537289, 11.26781369456864437475101337467

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