Properties

Label 2-624-156.59-c0-0-1
Degree $2$
Conductor $624$
Sign $0.999 - 0.0257i$
Analytic cond. $0.311416$
Root an. cond. $0.558047$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)3-s + (0.5 − 1.86i)7-s + (0.499 + 0.866i)9-s + (−0.866 + 0.5i)13-s + (−1.36 − 0.366i)19-s + (1.36 − 1.36i)21-s + i·25-s + 0.999i·27-s + (−0.366 + 0.366i)31-s + (0.366 + 1.36i)37-s − 0.999·39-s + (0.5 + 0.866i)43-s + (−2.36 − 1.36i)49-s + (−0.999 − i)57-s + (−0.866 − 1.5i)61-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)3-s + (0.5 − 1.86i)7-s + (0.499 + 0.866i)9-s + (−0.866 + 0.5i)13-s + (−1.36 − 0.366i)19-s + (1.36 − 1.36i)21-s + i·25-s + 0.999i·27-s + (−0.366 + 0.366i)31-s + (0.366 + 1.36i)37-s − 0.999·39-s + (0.5 + 0.866i)43-s + (−2.36 − 1.36i)49-s + (−0.999 − i)57-s + (−0.866 − 1.5i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0257i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0257i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $0.999 - 0.0257i$
Analytic conductor: \(0.311416\)
Root analytic conductor: \(0.558047\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (527, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :0),\ 0.999 - 0.0257i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.204667814\)
\(L(\frac12)\) \(\approx\) \(1.204667814\)
\(L(1)\) 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 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + (0.866 - 0.5i)T \)
good5 \( 1 - iT^{2} \)
7 \( 1 + (-0.5 + 1.86i)T + (-0.866 - 0.5i)T^{2} \)
11 \( 1 + (-0.866 + 0.5i)T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.366 - 0.366i)T - iT^{2} \)
37 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
41 \( 1 + (-0.866 + 0.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.866 + 0.5i)T^{2} \)
61 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.133 + 0.5i)T + (-0.866 + 0.5i)T^{2} \)
71 \( 1 + (-0.866 - 0.5i)T^{2} \)
73 \( 1 + (-0.366 + 0.366i)T - iT^{2} \)
79 \( 1 + iT - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (0.866 - 0.5i)T^{2} \)
97 \( 1 + (-0.5 + 1.86i)T + (-0.866 - 0.5i)T^{2} \)
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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

−10.69941595385522128318384696583, −9.985416981284775814364413091994, −9.177212072470621546888811377706, −8.104596583303451916414464294770, −7.44495704474233774776112083470, −6.65203453635774300029775452883, −4.81473958450904577445881581729, −4.33630652599484696112003490700, −3.26443349299923955959647998336, −1.75728726508139447476619980978, 2.10573740431087150174802393842, 2.65228169539454310578836083219, 4.18878779534206885630759197950, 5.46632740096823570222418438544, 6.29004420484763141937183977426, 7.50241911370540350028630008497, 8.337660594498790660203130587570, 8.879638004090138627527230885643, 9.698064287911136603158129280533, 10.80488735607858675521618633442

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