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Properties

Label 2-624-13.4-c3-0-13
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: $\chi_{624} (433, \cdot )$
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

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

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