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Properties

Label 2-1008-63.25-c1-0-31
Degree $2$
Conductor $1008$
Sign $-0.381 + 0.924i$
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  + (1.29 − 1.15i)3-s + (−1.84 + 3.20i)5-s + (−2.64 − 0.0963i)7-s + (0.349 − 2.97i)9-s + (−0.738 − 1.27i)11-s + (−1.34 − 2.33i)13-s + (1.29 + 6.27i)15-s + (3.28 − 5.69i)17-s + (0.444 + 0.769i)19-s + (−3.53 + 2.91i)21-s + (3.14 − 5.44i)23-s + (−4.34 − 7.52i)25-s + (−2.97 − 4.25i)27-s + (1.25 − 2.17i)29-s − 6.81·31-s + ⋯
L(s)  = 1  + (0.747 − 0.664i)3-s + (−0.827 + 1.43i)5-s + (−0.999 − 0.0364i)7-s + (0.116 − 0.993i)9-s + (−0.222 − 0.385i)11-s + (−0.374 − 0.648i)13-s + (0.334 + 1.62i)15-s + (0.797 − 1.38i)17-s + (0.101 + 0.176i)19-s + (−0.770 + 0.636i)21-s + (0.655 − 1.13i)23-s + (−0.868 − 1.50i)25-s + (−0.572 − 0.819i)27-s + (0.233 − 0.403i)29-s − 1.22·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.381 + 0.924i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.381 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.381 + 0.924i$
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: \((2,\ 1008,\ (\ :1/2),\ -0.381 + 0.924i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.006535138\)
\(L(\frac12)\) \(\approx\) \(1.006535138\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.29 + 1.15i)T \)
7 \( 1 + (2.64 + 0.0963i)T \)
good5 \( 1 + (1.84 - 3.20i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.738 + 1.27i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.34 + 2.33i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.28 + 5.69i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.444 - 0.769i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.14 + 5.44i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.25 + 2.17i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.81T + 31T^{2} \)
37 \( 1 + (1.38 + 2.40i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.05 + 3.56i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.00618 - 0.0107i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 6.98T + 47T^{2} \)
53 \( 1 + (1.60 - 2.78i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 6.90T + 59T^{2} \)
61 \( 1 + 5.73T + 61T^{2} \)
67 \( 1 - 9.46T + 67T^{2} \)
71 \( 1 - 5.46T + 71T^{2} \)
73 \( 1 + (6.03 - 10.4i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 11.4T + 79T^{2} \)
83 \( 1 + (2.23 - 3.87i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.43 + 7.68i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.58 - 11.4i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.733226935688596265543983745532, −8.811047444924023724113933697315, −7.75808409162752832234766893325, −7.23926742621640324071937594731, −6.66986301872846544429343174502, −5.61662349915611926324369840965, −3.92744546784520735714179940575, −3.02278571963472625523687047556, −2.68894725055064871122308063923, −0.41518756409511678105229080143, 1.60591048617870841063808884036, 3.24081505021332809111243316001, 3.95875791062765499669106217752, 4.82009077992418943764142267683, 5.67206903527121273293437546107, 7.14907381079103551712722124879, 7.889876101639626011039997130211, 8.730123339048630712061452357028, 9.297132172704084302141284371445, 9.935610789240334350177894391305

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