Properties

Label 2-624-13.10-c3-0-16
Degree $2$
Conductor $624$
Sign $0.711 + 0.702i$
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  + (−1.5 − 2.59i)3-s + 5.19i·5-s + (−9 − 5.19i)7-s + (−4.5 + 7.79i)9-s + (−45 + 25.9i)11-s + (−32.5 + 33.7i)13-s + (13.5 − 7.79i)15-s + (58.5 − 101. i)17-s + (21 + 12.1i)19-s + 31.1i·21-s + (9 + 15.5i)23-s + 98·25-s + 27·27-s + (49.5 + 85.7i)29-s − 193. i·31-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + 0.464i·5-s + (−0.485 − 0.280i)7-s + (−0.166 + 0.288i)9-s + (−1.23 + 0.712i)11-s + (−0.693 + 0.720i)13-s + (0.232 − 0.134i)15-s + (0.834 − 1.44i)17-s + (0.253 + 0.146i)19-s + 0.323i·21-s + (0.0815 + 0.141i)23-s + 0.784·25-s + 0.192·27-s + (0.316 + 0.548i)29-s − 1.12i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $0.711 + 0.702i$
Analytic conductor: \(36.8171\)
Root analytic conductor: \(6.06771\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :3/2),\ 0.711 + 0.702i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.229565595\)
\(L(\frac12)\) \(\approx\) \(1.229565595\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.5 + 2.59i)T \)
13 \( 1 + (32.5 - 33.7i)T \)
good5 \( 1 - 5.19iT - 125T^{2} \)
7 \( 1 + (9 + 5.19i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (45 - 25.9i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (-58.5 + 101. i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-21 - 12.1i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-9 - 15.5i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-49.5 - 85.7i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 193. iT - 2.97e4T^{2} \)
37 \( 1 + (-97.5 + 56.2i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (31.5 - 18.1i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (41 - 71.0i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 72.7iT - 1.03e5T^{2} \)
53 \( 1 + 261T + 1.48e5T^{2} \)
59 \( 1 + (-684 - 394. i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-359.5 + 622. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-609 + 351. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-405 - 233. i)T + (1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 - 684. iT - 3.89e5T^{2} \)
79 \( 1 - 440T + 4.93e5T^{2} \)
83 \( 1 + 1.19e3iT - 5.71e5T^{2} \)
89 \( 1 + (-1.31e3 + 758. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (1.00e3 + 578. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−9.978054246311598262815684125352, −9.545536924990670463170219257098, −8.082136654031600065174517999545, −7.24658401049696281176947893257, −6.80771346164178141153260604632, −5.49064688173042492403773877918, −4.69712234304784960951449528468, −3.14179778991720255905564109631, −2.22764543094477287694961305518, −0.53185232998327230977231932157, 0.790184876162296651817113842163, 2.66209880114558223685616071621, 3.61319423324662773162972473713, 5.02596931636481159153830956279, 5.53266283158403716782740069701, 6.56978221886597015938327675212, 7.942318623864111469084977677969, 8.476492507194088429748778599061, 9.608333577932919263002498478727, 10.34374121828936128909011264142

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