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Properties

Label 2-1008-63.25-c1-0-13
Degree $2$
Conductor $1008$
Sign $0.909 - 0.415i$
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  + (0.633 − 1.61i)3-s + (−1.70 + 2.95i)5-s + (0.410 − 2.61i)7-s + (−2.19 − 2.04i)9-s + (2.69 + 4.67i)11-s + (1.89 + 3.28i)13-s + (3.67 + 4.61i)15-s + (0.411 − 0.713i)17-s + (−0.233 − 0.404i)19-s + (−3.95 − 2.31i)21-s + (−2.74 + 4.76i)23-s + (−3.30 − 5.72i)25-s + (−4.68 + 2.25i)27-s + (0.400 − 0.693i)29-s + 9.90·31-s + ⋯
L(s)  = 1  + (0.365 − 0.930i)3-s + (−0.761 + 1.31i)5-s + (0.155 − 0.987i)7-s + (−0.732 − 0.680i)9-s + (0.813 + 1.40i)11-s + (0.525 + 0.910i)13-s + (0.949 + 1.19i)15-s + (0.0999 − 0.173i)17-s + (−0.0535 − 0.0928i)19-s + (−0.862 − 0.505i)21-s + (−0.573 + 0.993i)23-s + (−0.661 − 1.14i)25-s + (−0.901 + 0.433i)27-s + (0.0743 − 0.128i)29-s + 1.77·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 - 0.415i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.909 - 0.415i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.909 - 0.415i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.909 - 0.415i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.570328983\)
\(L(\frac12)\) \(\approx\) \(1.570328983\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.633 + 1.61i)T \)
7 \( 1 + (-0.410 + 2.61i)T \)
good5 \( 1 + (1.70 - 2.95i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.69 - 4.67i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.89 - 3.28i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.411 + 0.713i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.233 + 0.404i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.74 - 4.76i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.400 + 0.693i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 9.90T + 31T^{2} \)
37 \( 1 + (-4.34 - 7.52i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.84 - 3.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.36 + 7.55i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 10.4T + 47T^{2} \)
53 \( 1 + (4.71 - 8.17i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 1.66T + 59T^{2} \)
61 \( 1 - 0.948T + 61T^{2} \)
67 \( 1 + 0.539T + 67T^{2} \)
71 \( 1 - 3.86T + 71T^{2} \)
73 \( 1 + (-2.58 + 4.48i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 7.82T + 79T^{2} \)
83 \( 1 + (-3.79 + 6.57i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.73 + 6.46i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.22 - 5.58i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.03930991222633818222092383207, −9.209219210521939847975279736156, −7.974696540588276675008593877085, −7.43224577555895981048777736631, −6.80653326133091624306592952188, −6.28207065776912438069699985754, −4.38734481239672535980959707420, −3.74248179436909229047229207760, −2.58262856575631992696344767936, −1.33015451542862797955473929471, 0.793256854168056067204121963062, 2.69734619976201779615455291157, 3.78272868082244175233768677888, 4.50548727669319683374896394877, 5.53852368611413845886932443622, 6.10759082585828306509176515794, 8.029625672539673407586008430747, 8.390061706195291200940710927889, 8.872168357402453698123048631211, 9.680061285566860162037520256640

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