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Properties

Label 2-624-12.11-c1-0-19
Degree $2$
Conductor $624$
Sign $0.288 + 0.957i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.288 + 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.288 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $0.288 + 0.957i$
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: \((2,\ 624,\ (\ :1/2),\ 0.288 + 0.957i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55045 - 1.15191i\)
\(L(\frac12)\) \(\approx\) \(1.55045 - 1.15191i\)
\(L(\frac{3}{2})\) 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

−10.09920704256105819588692148907, −9.434048195847411212697806793580, −8.686557079570847552777824151813, −8.020331613622378950772278063616, −7.24963890630477088836088127458, −5.46345576385845490958311391923, −4.85741434924786195299521744300, −3.89087411357269691415642591505, −2.49392709744532629542843667777, −0.998652055157558217821855305999, 2.14075267968000683234012522401, 2.96095095970147335810817251500, 3.74661122153312623656375959767, 5.40946060418822697320645827342, 6.71304745696327545238791565132, 7.11406834479207167726061041300, 8.139292041555949093438198548464, 8.903042200006786262814964475698, 10.22613752134461315833807996046, 10.51547565226397172730469621057

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