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Properties

Label 2-1008-63.25-c1-0-21
Degree $2$
Conductor $1008$
Sign $0.254 + 0.967i$
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  + (−0.987 − 1.42i)3-s + (−1.38 + 2.40i)5-s + (−1.74 + 1.99i)7-s + (−1.04 + 2.81i)9-s + (−1.71 − 2.97i)11-s + (−0.429 − 0.743i)13-s + (4.78 − 0.398i)15-s + (−0.405 + 0.701i)17-s + (−0.750 − 1.29i)19-s + (4.55 + 0.508i)21-s + (3.82 − 6.62i)23-s + (−1.34 − 2.32i)25-s + (5.03 − 1.28i)27-s + (3.99 − 6.92i)29-s + 7.21·31-s + ⋯
L(s)  = 1  + (−0.570 − 0.821i)3-s + (−0.619 + 1.07i)5-s + (−0.657 + 0.753i)7-s + (−0.349 + 0.936i)9-s + (−0.518 − 0.898i)11-s + (−0.119 − 0.206i)13-s + (1.23 − 0.102i)15-s + (−0.0982 + 0.170i)17-s + (−0.172 − 0.298i)19-s + (0.993 + 0.110i)21-s + (0.797 − 1.38i)23-s + (−0.268 − 0.464i)25-s + (0.969 − 0.246i)27-s + (0.742 − 1.28i)29-s + 1.29·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.254 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.254 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.254 + 0.967i$
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: \((2,\ 1008,\ (\ :1/2),\ 0.254 + 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7207070766\)
\(L(\frac12)\) \(\approx\) \(0.7207070766\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.987 + 1.42i)T \)
7 \( 1 + (1.74 - 1.99i)T \)
good5 \( 1 + (1.38 - 2.40i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.71 + 2.97i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.429 + 0.743i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.405 - 0.701i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.750 + 1.29i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.82 + 6.62i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.99 + 6.92i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 7.21T + 31T^{2} \)
37 \( 1 + (-0.458 - 0.793i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.67 - 2.90i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.20 - 2.08i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 0.615T + 47T^{2} \)
53 \( 1 + (-6.31 + 10.9i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 1.46T + 59T^{2} \)
61 \( 1 - 11.4T + 61T^{2} \)
67 \( 1 + 16.2T + 67T^{2} \)
71 \( 1 + 14.4T + 71T^{2} \)
73 \( 1 + (4.16 - 7.22i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 2.75T + 79T^{2} \)
83 \( 1 + (-5.75 + 9.97i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.11 + 8.85i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.82 + 6.63i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.10364598581492224657161676281, −8.652463728002576962010631316660, −8.116144322163787206994249166501, −7.08101925353456411336135338376, −6.42996853062715538767275316987, −5.81105717675579755342436266912, −4.60083641477395731381615961452, −3.03248736580602691449243774429, −2.55066809852868445420398138043, −0.44418278579644101080997857855, 1.01861303791735459251611386423, 3.11101838532141716152763310962, 4.20329589055099731509745892832, 4.72469346524535856901861237745, 5.61218319446438756064513116914, 6.81760591156866104184869269638, 7.56235505355507990704988487927, 8.706100275260953318170332189256, 9.367478685778165100441582885568, 10.17858691584288245169142516303

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