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Properties

Label 2-624-12.11-c1-0-17
Degree $2$
Conductor $624$
Sign $-0.684 + 0.728i$
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  + (−1.68 − 0.396i)3-s − 4.25i·5-s + 0.792i·7-s + (2.68 + 1.33i)9-s + 4·11-s + 13-s + (−1.68 + 7.17i)15-s − 5.84i·17-s − 3.46i·19-s + (0.313 − 1.33i)21-s − 6.74·23-s − 13.1·25-s + (−4 − 3.31i)27-s + 6.63i·31-s + (−6.74 − 1.58i)33-s + ⋯
L(s)  = 1  + (−0.973 − 0.228i)3-s − 1.90i·5-s + 0.299i·7-s + (0.895 + 0.445i)9-s + 1.20·11-s + 0.277·13-s + (−0.435 + 1.85i)15-s − 1.41i·17-s − 0.794i·19-s + (0.0684 − 0.291i)21-s − 1.40·23-s − 2.62·25-s + (−0.769 − 0.638i)27-s + 1.19i·31-s + (−1.17 − 0.275i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.684 + 0.728i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.684 + 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $-0.684 + 0.728i$
Analytic conductor: \(4.98266\)
Root analytic conductor: \(2.23218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :1/2),\ -0.684 + 0.728i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.371315 - 0.858498i\)
\(L(\frac12)\) \(\approx\) \(0.371315 - 0.858498i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.68 + 0.396i)T \)
13 \( 1 - T \)
good5 \( 1 + 4.25iT - 5T^{2} \)
7 \( 1 - 0.792iT - 7T^{2} \)
11 \( 1 - 4T + 11T^{2} \)
17 \( 1 + 5.84iT - 17T^{2} \)
19 \( 1 + 3.46iT - 19T^{2} \)
23 \( 1 + 6.74T + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 6.63iT - 31T^{2} \)
37 \( 1 + 5.37T + 37T^{2} \)
41 \( 1 + 1.58iT - 41T^{2} \)
43 \( 1 + 9.30iT - 43T^{2} \)
47 \( 1 + 0.627T + 47T^{2} \)
53 \( 1 - 5.34iT - 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 - 11.4T + 61T^{2} \)
67 \( 1 + 3.46iT - 67T^{2} \)
71 \( 1 + 6.11T + 71T^{2} \)
73 \( 1 - 0.744T + 73T^{2} \)
79 \( 1 + 5.04iT - 79T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 - 1.58iT - 89T^{2} \)
97 \( 1 - 8.74T + 97T^{2} \)
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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

−10.25187730317615105435863346053, −9.199639596949058485443482259286, −8.818181034392254124449879815926, −7.60676360611921874100030804203, −6.56325200111092704837609418701, −5.53434943102266623332198932671, −4.88792639861953490308290599740, −4.00309779132648861517238934406, −1.75436842134219052060051738825, −0.60536952867606288147183947334, 1.82144672924632679991385996686, 3.60385773305225219995500834002, 4.09223484256082060740310952795, 6.03127486472029471416356982318, 6.20792102683614939103840071986, 7.13478197469516075853474839665, 8.085206658098904541675798132115, 9.658298748263830257124337250619, 10.23615311630885182979299347909, 10.89289428309199668402881831112

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