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Properties

Label 2-504-63.25-c1-0-7
Degree $2$
Conductor $504$
Sign $0.927 - 0.373i$
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  + (−0.704 − 1.58i)3-s + (−1.05 + 1.82i)5-s + (2.58 + 0.569i)7-s + (−2.00 + 2.22i)9-s + (−0.199 − 0.345i)11-s + (1.44 + 2.49i)13-s + (3.62 + 0.381i)15-s + (−0.176 + 0.305i)17-s + (2.84 + 4.93i)19-s + (−0.918 − 4.48i)21-s + (0.438 − 0.759i)23-s + (0.285 + 0.494i)25-s + (4.94 + 1.60i)27-s + (0.874 − 1.51i)29-s + 9.13·31-s + ⋯
L(s)  = 1  + (−0.406 − 0.913i)3-s + (−0.470 + 0.815i)5-s + (0.976 + 0.215i)7-s + (−0.669 + 0.742i)9-s + (−0.0601 − 0.104i)11-s + (0.400 + 0.693i)13-s + (0.935 + 0.0985i)15-s + (−0.0428 + 0.0741i)17-s + (0.653 + 1.13i)19-s + (−0.200 − 0.979i)21-s + (0.0914 − 0.158i)23-s + (0.0571 + 0.0989i)25-s + (0.950 + 0.309i)27-s + (0.162 − 0.281i)29-s + 1.64·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.373i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 - 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.927 - 0.373i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.927 - 0.373i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20169 + 0.232746i\)
\(L(\frac12)\) \(\approx\) \(1.20169 + 0.232746i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.704 + 1.58i)T \)
7 \( 1 + (-2.58 - 0.569i)T \)
good5 \( 1 + (1.05 - 1.82i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.199 + 0.345i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.44 - 2.49i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.176 - 0.305i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.84 - 4.93i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.438 + 0.759i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.874 + 1.51i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 9.13T + 31T^{2} \)
37 \( 1 + (3.39 + 5.88i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.20 - 2.08i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.276 + 0.479i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 11.7T + 47T^{2} \)
53 \( 1 + (2.07 - 3.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 9.32T + 59T^{2} \)
61 \( 1 + 10.0T + 61T^{2} \)
67 \( 1 - 1.20T + 67T^{2} \)
71 \( 1 + 14.6T + 71T^{2} \)
73 \( 1 + (-0.315 + 0.546i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 2.48T + 79T^{2} \)
83 \( 1 + (4.59 - 7.95i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-7.29 - 12.6i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.84 - 13.5i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.07475538280238756199136207909, −10.47525106482517114241383498449, −8.986794881715451921573520288603, −7.995668884523080384325576143282, −7.43057760947942663921122847772, −6.43040832750939331573752929043, −5.55129845695224195665284361170, −4.26874308023444363022729965705, −2.79417628857429253375282778592, −1.45198215798705696256194990616, 0.894818482215425087890803388966, 3.08306654967651023464948120985, 4.49116871127694709419609374471, 4.86556248962259874934301774985, 5.95920157470506688161269103597, 7.36649091081233559652332201479, 8.410855588874371812928344041307, 8.965361516434545202064567977397, 10.12650594686346653759708610427, 10.85769396553242142644352491759

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