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Properties

Label 2-624-52.31-c3-0-34
Degree $2$
Conductor $624$
Sign $-0.0655 + 0.997i$
Analytic cond. $36.8171$
Root an. cond. $6.06771$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3i·3-s + (13.7 − 13.7i)5-s + (−9.00 + 9.00i)7-s − 9·9-s + (5.13 − 5.13i)11-s + (−29.7 − 36.2i)13-s + (41.1 + 41.1i)15-s − 41.0i·17-s + (28.6 + 28.6i)19-s + (−27.0 − 27.0i)21-s + 58.8·23-s − 251. i·25-s − 27i·27-s − 13.6·29-s + (−163. − 163. i)31-s + ⋯
L(s)  = 1  + 0.577i·3-s + (1.22 − 1.22i)5-s + (−0.486 + 0.486i)7-s − 0.333·9-s + (0.140 − 0.140i)11-s + (−0.634 − 0.772i)13-s + (0.708 + 0.708i)15-s − 0.586i·17-s + (0.345 + 0.345i)19-s + (−0.280 − 0.280i)21-s + 0.533·23-s − 2.01i·25-s − 0.192i·27-s − 0.0871·29-s + (−0.949 − 0.949i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0655 + 0.997i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0655 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $-0.0655 + 0.997i$
Analytic conductor: \(36.8171\)
Root analytic conductor: \(6.06771\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :3/2),\ -0.0655 + 0.997i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.691037334\)
\(L(\frac12)\) \(\approx\) \(1.691037334\)
\(L(\frac{5}{2})\) 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 \( 1 - 3iT \)
13 \( 1 + (29.7 + 36.2i)T \)
good5 \( 1 + (-13.7 + 13.7i)T - 125iT^{2} \)
7 \( 1 + (9.00 - 9.00i)T - 343iT^{2} \)
11 \( 1 + (-5.13 + 5.13i)T - 1.33e3iT^{2} \)
17 \( 1 + 41.0iT - 4.91e3T^{2} \)
19 \( 1 + (-28.6 - 28.6i)T + 6.85e3iT^{2} \)
23 \( 1 - 58.8T + 1.21e4T^{2} \)
29 \( 1 + 13.6T + 2.43e4T^{2} \)
31 \( 1 + (163. + 163. i)T + 2.97e4iT^{2} \)
37 \( 1 + (129. + 129. i)T + 5.06e4iT^{2} \)
41 \( 1 + (-30.4 + 30.4i)T - 6.89e4iT^{2} \)
43 \( 1 + 342.T + 7.95e4T^{2} \)
47 \( 1 + (-328. + 328. i)T - 1.03e5iT^{2} \)
53 \( 1 - 109.T + 1.48e5T^{2} \)
59 \( 1 + (262. - 262. i)T - 2.05e5iT^{2} \)
61 \( 1 - 69.0T + 2.26e5T^{2} \)
67 \( 1 + (138. + 138. i)T + 3.00e5iT^{2} \)
71 \( 1 + (185. + 185. i)T + 3.57e5iT^{2} \)
73 \( 1 + (597. + 597. i)T + 3.89e5iT^{2} \)
79 \( 1 + 627. iT - 4.93e5T^{2} \)
83 \( 1 + (-542. - 542. i)T + 5.71e5iT^{2} \)
89 \( 1 + (306. + 306. i)T + 7.04e5iT^{2} \)
97 \( 1 + (-1.22e3 + 1.22e3i)T - 9.12e5iT^{2} \)
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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

−9.792640400056007369723579143549, −9.223086490333367589612988149037, −8.616461600497029111811813273266, −7.35519398465778891437753888925, −5.92096944728621968564785364345, −5.46906053493843936499865424707, −4.60657004335798388775960398231, −3.16060199660410751102279785536, −1.96147963979710766711543701854, −0.46223584914440797692589799938, 1.50378968500484842761152958879, 2.51898618449088654000371279958, 3.52914279873816417128053973270, 5.11647426442156891306982492757, 6.21858057099884688307364656625, 6.85227851393895268209196009888, 7.37941451074813998292974319321, 8.858569986799420875549055404987, 9.691786072537563676170887815592, 10.38606883862940723633403030941

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