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Properties

Label 2-624-52.31-c3-0-37
Degree $2$
Conductor $624$
Sign $0.318 + 0.947i$
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 + (7.87 − 7.87i)5-s + (15.6 − 15.6i)7-s − 9·9-s + (35.2 − 35.2i)11-s + (39.8 − 24.6i)13-s + (23.6 + 23.6i)15-s − 8.37i·17-s + (−3.94 − 3.94i)19-s + (46.8 + 46.8i)21-s − 192.·23-s + 0.965i·25-s − 27i·27-s − 8.58·29-s + (−214. − 214. i)31-s + ⋯
L(s)  = 1  + 0.577i·3-s + (0.704 − 0.704i)5-s + (0.843 − 0.843i)7-s − 0.333·9-s + (0.965 − 0.965i)11-s + (0.850 − 0.526i)13-s + (0.406 + 0.406i)15-s − 0.119i·17-s + (−0.0475 − 0.0475i)19-s + (0.487 + 0.487i)21-s − 1.74·23-s + 0.00772i·25-s − 0.192i·27-s − 0.0549·29-s + (−1.24 − 1.24i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.318 + 0.947i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.318 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $0.318 + 0.947i$
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: \((2,\ 624,\ (\ :3/2),\ 0.318 + 0.947i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.464675413\)
\(L(\frac12)\) \(\approx\) \(2.464675413\)
\(L(\frac{5}{2})\) 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 \( 1 + (-39.8 + 24.6i)T \)
good5 \( 1 + (-7.87 + 7.87i)T - 125iT^{2} \)
7 \( 1 + (-15.6 + 15.6i)T - 343iT^{2} \)
11 \( 1 + (-35.2 + 35.2i)T - 1.33e3iT^{2} \)
17 \( 1 + 8.37iT - 4.91e3T^{2} \)
19 \( 1 + (3.94 + 3.94i)T + 6.85e3iT^{2} \)
23 \( 1 + 192.T + 1.21e4T^{2} \)
29 \( 1 + 8.58T + 2.43e4T^{2} \)
31 \( 1 + (214. + 214. i)T + 2.97e4iT^{2} \)
37 \( 1 + (86.7 + 86.7i)T + 5.06e4iT^{2} \)
41 \( 1 + (-210. + 210. i)T - 6.89e4iT^{2} \)
43 \( 1 + 23.0T + 7.95e4T^{2} \)
47 \( 1 + (201. - 201. i)T - 1.03e5iT^{2} \)
53 \( 1 - 73.3T + 1.48e5T^{2} \)
59 \( 1 + (240. - 240. i)T - 2.05e5iT^{2} \)
61 \( 1 + 244.T + 2.26e5T^{2} \)
67 \( 1 + (-579. - 579. i)T + 3.00e5iT^{2} \)
71 \( 1 + (-15.1 - 15.1i)T + 3.57e5iT^{2} \)
73 \( 1 + (-564. - 564. i)T + 3.89e5iT^{2} \)
79 \( 1 + 507. iT - 4.93e5T^{2} \)
83 \( 1 + (316. + 316. i)T + 5.71e5iT^{2} \)
89 \( 1 + (-818. - 818. i)T + 7.04e5iT^{2} \)
97 \( 1 + (-1.16e3 + 1.16e3i)T - 9.12e5iT^{2} \)
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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

−10.00215785987538787860157058404, −9.157068500924300488071836104844, −8.448344666479693887598396534661, −7.56010662115008338358008736913, −6.09065127909394839319091276359, −5.54392059653323310936959646017, −4.30014345873142999747276268523, −3.62643098060382075334988131407, −1.80657922466578258315293002602, −0.71433434350634372443198604577, 1.65338381517421663343594886386, 2.10996391767897053127233782991, 3.67328236189005273401370597405, 4.97068404970793203531783879776, 6.13789303149477211059804738951, 6.59548677962459249981839852383, 7.72616829392497531040220029266, 8.647262836880531603493320136038, 9.442797929210295860039247010727, 10.40206786912738959308233802496

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