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Properties

Label 2-624-39.20-c1-0-16
Degree $2$
Conductor $624$
Sign $0.265 + 0.964i$
Analytic cond. $4.98266$
Root an. cond. $2.23218$
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  + (−0.866 + 1.5i)3-s + (−2.23 − 0.598i)7-s + (−1.5 − 2.59i)9-s + (−0.866 − 3.5i)13-s + (2.09 − 7.83i)19-s + (2.83 − 2.83i)21-s + 5i·25-s + 5.19·27-s + (−4.63 − 4.63i)31-s + (−0.562 − 2.09i)37-s + (6 + 1.73i)39-s + (10.5 − 6.06i)43-s + (−1.43 − 0.830i)49-s + (9.92 + 9.92i)57-s + (−7.79 − 13.5i)61-s + ⋯
L(s)  = 1  + (−0.499 + 0.866i)3-s + (−0.843 − 0.226i)7-s + (−0.5 − 0.866i)9-s + (−0.240 − 0.970i)13-s + (0.481 − 1.79i)19-s + (0.617 − 0.617i)21-s + i·25-s + 1.00·27-s + (−0.832 − 0.832i)31-s + (−0.0924 − 0.344i)37-s + (0.960 + 0.277i)39-s + (1.60 − 0.924i)43-s + (−0.205 − 0.118i)49-s + (1.31 + 1.31i)57-s + (−0.997 − 1.72i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.265 + 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.265 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $0.265 + 0.964i$
Analytic conductor: \(4.98266\)
Root analytic conductor: \(2.23218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :1/2),\ 0.265 + 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.549820 - 0.419036i\)
\(L(\frac12)\) \(\approx\) \(0.549820 - 0.419036i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 - 1.5i)T \)
13 \( 1 + (0.866 + 3.5i)T \)
good5 \( 1 - 5iT^{2} \)
7 \( 1 + (2.23 + 0.598i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.09 + 7.83i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.63 + 4.63i)T + 31iT^{2} \)
37 \( 1 + (0.562 + 2.09i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-10.5 + 6.06i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 47iT^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (7.79 + 13.5i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (14.0 - 3.76i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-11.2 + 11.2i)T - 73iT^{2} \)
79 \( 1 + 5.19T + 79T^{2} \)
83 \( 1 - 83iT^{2} \)
89 \( 1 + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (4.42 - 16.5i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.55981380344716680920078389812, −9.405181089378338595567384925369, −9.195975935129912992123978044009, −7.69102523517801316541474215193, −6.76771312757655289382450785896, −5.73089768310786501243845979791, −4.95284056954874525983196800956, −3.75781921212278492290750664748, −2.84601169693423664069933231679, −0.40656572516366267230471561882, 1.54553894668045087705020238489, 2.87968410847943133397022746025, 4.27416567008580165361832457270, 5.62707708604292567939862738929, 6.28446151216042068908027111106, 7.14894865048505321178731596140, 8.012130020638808137508393085167, 9.044558919876929694698391058381, 9.975578000569516163423422145530, 10.83167512686094085021240608750

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