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Properties

Label 2-624-156.11-c0-0-0
Degree $2$
Conductor $624$
Sign $0.846 - 0.533i$
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 − 0.133i)7-s + (0.499 − 0.866i)9-s + (0.866 + 0.5i)13-s + (0.366 + 1.36i)19-s + (−0.366 + 0.366i)21-s + i·25-s + 0.999i·27-s + (1.36 − 1.36i)31-s + (−1.36 − 0.366i)37-s − 0.999·39-s + (0.5 − 0.866i)43-s + (−0.633 + 0.366i)49-s + (−1 − 0.999i)57-s + (0.866 − 1.5i)61-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)3-s + (0.5 − 0.133i)7-s + (0.499 − 0.866i)9-s + (0.866 + 0.5i)13-s + (0.366 + 1.36i)19-s + (−0.366 + 0.366i)21-s + i·25-s + 0.999i·27-s + (1.36 − 1.36i)31-s + (−1.36 − 0.366i)37-s − 0.999·39-s + (0.5 − 0.866i)43-s + (−0.633 + 0.366i)49-s + (−1 − 0.999i)57-s + (0.866 − 1.5i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.846 - 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.846 - 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $0.846 - 0.533i$
Analytic conductor: \(0.311416\)
Root analytic conductor: \(0.558047\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (479, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :0),\ 0.846 - 0.533i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7751850948\)
\(L(\frac12)\) \(\approx\) \(0.7751850948\)
\(L(1)\) 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 \( 1 + (-0.5 + 0.133i)T + (0.866 - 0.5i)T^{2} \)
11 \( 1 + (0.866 + 0.5i)T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-1.36 + 1.36i)T - iT^{2} \)
37 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
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 \( 1 + (1.86 + 0.5i)T + (0.866 + 0.5i)T^{2} \)
71 \( 1 + (0.866 - 0.5i)T^{2} \)
73 \( 1 + (1.36 - 1.36i)T - iT^{2} \)
79 \( 1 + iT - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (-0.866 - 0.5i)T^{2} \)
97 \( 1 + (-0.5 + 0.133i)T + (0.866 - 0.5i)T^{2} \)
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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

−10.93737569963180392781951689403, −10.15542526384730755915782672623, −9.324869657790859895490548987205, −8.316136226045793589950036414262, −7.28125939559446381453837028945, −6.21145106005320949398561038009, −5.48792227476808135320663056813, −4.39560614751817762287344865172, −3.53422061155349066284012715185, −1.52423635579866441641838414458, 1.25765841056284951271193415819, 2.83124371596692076575502481150, 4.47873119961474167082152471421, 5.27630608086848846561698657734, 6.28496426841849807117411681453, 7.04877925519801534311121349280, 8.098128798354687318181897452188, 8.829861538094837705689115028047, 10.20157644483050137758686147256, 10.78019915671042769974118696045

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