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Properties

Label 2-504-63.58-c1-0-10
Degree $2$
Conductor $504$
Sign $0.294 + 0.955i$
Analytic cond. $4.02446$
Root an. cond. $2.00610$
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  + (−1.65 − 0.523i)3-s + (−0.841 − 1.45i)5-s + (1.65 + 2.06i)7-s + (2.45 + 1.72i)9-s + (−0.622 + 1.07i)11-s + (1.96 − 3.39i)13-s + (0.626 + 2.84i)15-s + (−1.62 − 2.81i)17-s + (2.36 − 4.09i)19-s + (−1.65 − 4.27i)21-s + (0.199 + 0.344i)23-s + (1.08 − 1.87i)25-s + (−3.14 − 4.13i)27-s + (−3.19 − 5.54i)29-s − 0.578·31-s + ⋯
L(s)  = 1  + (−0.953 − 0.302i)3-s + (−0.376 − 0.651i)5-s + (0.625 + 0.780i)7-s + (0.817 + 0.576i)9-s + (−0.187 + 0.325i)11-s + (0.543 − 0.941i)13-s + (0.161 + 0.735i)15-s + (−0.394 − 0.683i)17-s + (0.541 − 0.938i)19-s + (−0.360 − 0.932i)21-s + (0.0415 + 0.0718i)23-s + (0.216 − 0.375i)25-s + (−0.604 − 0.796i)27-s + (−0.594 − 1.02i)29-s − 0.103·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.294 + 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.294 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.294 + 0.955i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.294 + 0.955i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.779688 - 0.575338i\)
\(L(\frac12)\) \(\approx\) \(0.779688 - 0.575338i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.65 + 0.523i)T \)
7 \( 1 + (-1.65 - 2.06i)T \)
good5 \( 1 + (0.841 + 1.45i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.622 - 1.07i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.96 + 3.39i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.62 + 2.81i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.36 + 4.09i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.199 - 0.344i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.19 + 5.54i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 0.578T + 31T^{2} \)
37 \( 1 + (-2.72 + 4.71i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.20 + 7.27i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.46 - 4.26i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 0.425T + 47T^{2} \)
53 \( 1 + (0.466 + 0.807i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 6.05T + 59T^{2} \)
61 \( 1 - 10.2T + 61T^{2} \)
67 \( 1 + 9.41T + 67T^{2} \)
71 \( 1 - 8.46T + 71T^{2} \)
73 \( 1 + (-6.82 - 11.8i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 5.53T + 79T^{2} \)
83 \( 1 + (8.03 + 13.9i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (6.03 - 10.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.86 + 10.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.13964847501536052845923594188, −9.909017216838674930621341008450, −8.887932014539684168905828554504, −7.971567434603356229456123450386, −7.14808737396611332281141241671, −5.84459020586953351149653413480, −5.18987799422661449008420135559, −4.28942025281433834325958726840, −2.40625780442605696993435953406, −0.73841585416133661190646578768, 1.41651964553109203068790263629, 3.54753115521460238712110689257, 4.33497383575488485478498802825, 5.49826233590120266389478958190, 6.55436693430683961950249079657, 7.27792345417610486490677924642, 8.315438059145772213720786857482, 9.546298015547297928024611303920, 10.53418401183851071231571971299, 11.09370575229707788936836405950

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