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Properties

Label 2-1008-63.58-c1-0-12
Degree $2$
Conductor $1008$
Sign $0.964 + 0.262i$
Analytic cond. $8.04892$
Root an. cond. $2.83706$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.262i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 + 0.262i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.964 + 0.262i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (625, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.964 + 0.262i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.072732764\)
\(L(\frac12)\) \(\approx\) \(1.072732764\)
\(L(\frac{3}{2})\) 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

−10.23070812237892164639764897211, −8.788919534544587309139373230099, −8.236338197105008065509163206762, −7.57146109022385378779603514328, −6.32144662689006173621269742320, −5.69422413997251172668490863086, −4.97341887483639272168638670029, −3.62407396109202787745775841484, −2.27214822539353163513701939871, −0.934849982196174165786965915114, 0.75806167273809316065007213949, 2.98088058270800704184233906465, 3.76774369150015926409338146360, 4.62110420260357015130031112208, 5.71111645069640655938642506160, 6.71839190780547668043738630972, 7.18435727380486293873498017040, 8.413018492406314468682201770666, 9.430208859531889475971648029384, 10.05456265511147264236107887122

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