Properties

Label 2-624-156.119-c0-0-1
Degree $2$
Conductor $624$
Sign $0.477 + 0.878i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.477 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.477 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $0.477 + 0.878i$
Analytic conductor: \(0.311416\)
Root analytic conductor: \(0.558047\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :0),\ 0.477 + 0.878i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6108614497\)
\(L(\frac12)\) \(\approx\) \(0.6108614497\)
\(L(1)\) 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

−10.50166965958906157216421912121, −10.06149869451287955972074173062, −9.312780598764084002743101039789, −7.71070863978518065064309355484, −7.12178598663380236040742671321, −6.20307338343528435494364435340, −5.01352425793039458520971770938, −4.22009431994373522937465165834, −3.18693731212803509935053836816, −0.805185622971814295299179484791, 1.89311658766479411981328128539, 3.07251215638947684371872781925, 4.89395072277040478142646555029, 5.55192510052919265858584244269, 6.36457961974624249878133078678, 7.29989393595632467944713290454, 8.315205720545005785757717558074, 9.406752352808669447380759493125, 9.936398568455412265000168408052, 11.32803521643644268022226661697

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