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Properties

Label 2-504-63.25-c1-0-9
Degree $2$
Conductor $504$
Sign $0.964 - 0.262i$
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  + (1.12 − 1.31i)3-s + (−0.927 + 1.60i)5-s + (0.900 + 2.48i)7-s + (−0.467 − 2.96i)9-s + (1.28 + 2.23i)11-s + (2.82 + 4.88i)13-s + (1.07 + 3.03i)15-s + (3.57 − 6.19i)17-s + (0.636 + 1.10i)19-s + (4.28 + 1.61i)21-s + (−0.120 + 0.208i)23-s + (0.777 + 1.34i)25-s + (−4.42 − 2.71i)27-s + (0.923 − 1.59i)29-s − 2.99·31-s + ⋯
L(s)  = 1  + (0.649 − 0.760i)3-s + (−0.414 + 0.718i)5-s + (0.340 + 0.940i)7-s + (−0.155 − 0.987i)9-s + (0.388 + 0.672i)11-s + (0.782 + 1.35i)13-s + (0.276 + 0.782i)15-s + (0.868 − 1.50i)17-s + (0.146 + 0.252i)19-s + (0.935 + 0.352i)21-s + (−0.0251 + 0.0435i)23-s + (0.155 + 0.269i)25-s + (−0.852 − 0.523i)27-s + (0.171 − 0.297i)29-s − 0.537·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.262i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 - 0.262i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.964 - 0.262i$
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: \((2,\ 504,\ (\ :1/2),\ 0.964 - 0.262i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.75215 + 0.234112i\)
\(L(\frac12)\) \(\approx\) \(1.75215 + 0.234112i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.12 + 1.31i)T \)
7 \( 1 + (-0.900 - 2.48i)T \)
good5 \( 1 + (0.927 - 1.60i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.28 - 2.23i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.82 - 4.88i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.57 + 6.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.636 - 1.10i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.120 - 0.208i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.923 + 1.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 2.99T + 31T^{2} \)
37 \( 1 + (-0.338 - 0.585i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.733 + 1.27i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.14 + 7.17i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 12.3T + 47T^{2} \)
53 \( 1 + (-3.35 + 5.81i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 2.08T + 59T^{2} \)
61 \( 1 - 12.9T + 61T^{2} \)
67 \( 1 + 4.83T + 67T^{2} \)
71 \( 1 + 1.53T + 71T^{2} \)
73 \( 1 + (6.55 - 11.3i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 3.72T + 79T^{2} \)
83 \( 1 + (3.00 - 5.19i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-6.60 - 11.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.40 + 11.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.43475186157688280733333035561, −9.801091372269606252476349418993, −9.108684282521275105582422950378, −8.252409814563722680058724517106, −7.20735602710741199715680634789, −6.70877060377446884003497024578, −5.44156339728934977480975529631, −3.94629855738796173159843669238, −2.83956667379015788447968069829, −1.68471930028295832378873077915, 1.17613937084415072577626062055, 3.28233143980171057805989726883, 3.96081844130690263548489454620, 5.00638403784663616487903445787, 6.09063951941219854331557839107, 7.73242619410820330563144293522, 8.214335619987686290879329261729, 8.922852302815333811258858672980, 10.18169612957085126123905478621, 10.64866517362411496674173789593

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