Properties

Label 2-624-13.12-c1-0-11
Degree $2$
Conductor $624$
Sign $0.277 + 0.960i$
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  + 3-s − 3.46i·7-s + 9-s − 3.46i·11-s + (−1 − 3.46i)13-s − 6·17-s + 3.46i·19-s − 3.46i·21-s + 5·25-s + 27-s + 6·29-s − 3.46i·31-s − 3.46i·33-s − 6.92i·37-s + (−1 − 3.46i)39-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.30i·7-s + 0.333·9-s − 1.04i·11-s + (−0.277 − 0.960i)13-s − 1.45·17-s + 0.794i·19-s − 0.755i·21-s + 25-s + 0.192·27-s + 1.11·29-s − 0.622i·31-s − 0.603i·33-s − 1.13i·37-s + (−0.160 − 0.554i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $0.277 + 0.960i$
Analytic conductor: \(4.98266\)
Root analytic conductor: \(2.23218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :1/2),\ 0.277 + 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30885 - 0.984469i\)
\(L(\frac12)\) \(\approx\) \(1.30885 - 0.984469i\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
13 \( 1 + (1 + 3.46i)T \)
good5 \( 1 - 5T^{2} \)
7 \( 1 + 3.46iT - 7T^{2} \)
11 \( 1 + 3.46iT - 11T^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 - 3.46iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 3.46iT - 31T^{2} \)
37 \( 1 + 6.92iT - 37T^{2} \)
41 \( 1 - 6.92iT - 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 - 3.46iT - 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 - 10.3iT - 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 10.3iT - 67T^{2} \)
71 \( 1 - 3.46iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 3.46iT - 83T^{2} \)
89 \( 1 - 6.92iT - 89T^{2} \)
97 \( 1 - 13.8iT - 97T^{2} \)
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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

−10.66319121561289677400757188784, −9.562311045310686475041676405335, −8.576641639165694473556274004970, −7.86718136373754710577523334846, −6.99088465860776900368203948303, −6.00493265352164549466872659514, −4.64550640467252958945267459081, −3.73598920899971406734335314969, −2.66075473752642768838636405485, −0.860671078871469360885599871875, 2.00657706072832352684294701813, 2.75771868111859403138891802347, 4.36860633865166102426951112563, 5.07844470027483760381842639919, 6.56082946455080767729923637818, 7.08042949451288875981485634498, 8.510765237236881745588180510578, 8.935791168218723743976795167051, 9.686147721688605379665328424707, 10.75689756272077707582952716091

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