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Properties

Label 2-504-63.58-c1-0-18
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: $\chi_{504} (121, \cdot )$
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

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

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