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Properties

Label 2-1008-63.4-c1-0-17
Degree $2$
Conductor $1008$
Sign $0.0250 - 0.999i$
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  + (1.04 + 1.38i)3-s − 0.0619·5-s + (1.63 + 2.07i)7-s + (−0.828 + 2.88i)9-s + 3.18·11-s + (−0.252 + 0.437i)13-s + (−0.0645 − 0.0857i)15-s + (−0.554 + 0.960i)17-s + (−0.933 − 1.61i)19-s + (−1.16 + 4.43i)21-s + 6.20·23-s − 4.99·25-s + (−4.85 + 1.85i)27-s + (2.39 + 4.15i)29-s + (−1.26 − 2.19i)31-s + ⋯
L(s)  = 1  + (0.601 + 0.798i)3-s − 0.0277·5-s + (0.618 + 0.785i)7-s + (−0.276 + 0.961i)9-s + 0.958·11-s + (−0.0700 + 0.121i)13-s + (−0.0166 − 0.0221i)15-s + (−0.134 + 0.233i)17-s + (−0.214 − 0.370i)19-s + (−0.255 + 0.966i)21-s + 1.29·23-s − 0.999·25-s + (−0.933 + 0.357i)27-s + (0.445 + 0.770i)29-s + (−0.227 − 0.394i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0250 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0250 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.0250 - 0.999i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.0250 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.089164543\)
\(L(\frac12)\) \(\approx\) \(2.089164543\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.04 - 1.38i)T \)
7 \( 1 + (-1.63 - 2.07i)T \)
good5 \( 1 + 0.0619T + 5T^{2} \)
11 \( 1 - 3.18T + 11T^{2} \)
13 \( 1 + (0.252 - 0.437i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.554 - 0.960i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.933 + 1.61i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 6.20T + 23T^{2} \)
29 \( 1 + (-2.39 - 4.15i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.26 + 2.19i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.26 + 7.38i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.94 - 8.56i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.95 - 6.85i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.29 + 5.70i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.58 - 2.74i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.50 - 7.80i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.94 + 12.0i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.66 - 2.88i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.25T + 71T^{2} \)
73 \( 1 + (-2.07 + 3.59i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.48 - 2.57i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.17 + 3.76i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.30 + 7.44i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.27 + 5.67i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.02380365674880662250575324513, −9.148314471068252090847328946065, −8.771291882469054866572707757003, −7.88838828118364999487392697992, −6.84680198418993667792155497273, −5.69434466505733846864495351448, −4.83710918494925992616422848460, −3.97724021174665366468275921362, −2.88852053533693497121217944410, −1.75469789376859444361931807820, 0.962926997419872313344997594847, 2.05487940875498147879699281425, 3.41958301772535488863600818181, 4.25650018710597192861786640119, 5.51077803432249878988019462602, 6.71498939825637995220324707106, 7.16424261111893802724509639104, 8.119114324171206737362146005053, 8.758136768854883761928085335995, 9.647394500170620842703296692488

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