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Properties

Label 2-624-13.10-c1-0-10
Degree $2$
Conductor $624$
Sign $-0.252 + 0.967i$
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  + (−0.5 − 0.866i)3-s − 3.46i·5-s + (1.5 + 0.866i)7-s + (−0.499 + 0.866i)9-s + (3 − 1.73i)11-s + (3.5 + 0.866i)13-s + (−2.99 + 1.73i)15-s + (−3 − 1.73i)19-s − 1.73i·21-s + (−3 − 5.19i)23-s − 6.99·25-s + 0.999·27-s + (−3 − 5.19i)29-s + 1.73i·31-s + (−3 − 1.73i)33-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s − 1.54i·5-s + (0.566 + 0.327i)7-s + (−0.166 + 0.288i)9-s + (0.904 − 0.522i)11-s + (0.970 + 0.240i)13-s + (−0.774 + 0.447i)15-s + (−0.688 − 0.397i)19-s − 0.377i·21-s + (−0.625 − 1.08i)23-s − 1.39·25-s + 0.192·27-s + (−0.557 − 0.964i)29-s + 0.311i·31-s + (−0.522 − 0.301i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.252 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $-0.252 + 0.967i$
Analytic conductor: \(4.98266\)
Root analytic conductor: \(2.23218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :1/2),\ -0.252 + 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.859310 - 1.11247i\)
\(L(\frac12)\) \(\approx\) \(0.859310 - 1.11247i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-3.5 - 0.866i)T \)
good5 \( 1 + 3.46iT - 5T^{2} \)
7 \( 1 + (-1.5 - 0.866i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-3 + 1.73i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3 + 1.73i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.73iT - 31T^{2} \)
37 \( 1 + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (6 - 3.46i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 3.46iT - 47T^{2} \)
53 \( 1 - 12T + 53T^{2} \)
59 \( 1 + (-3 - 1.73i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.5 - 4.33i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-9 - 5.19i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 1.73iT - 73T^{2} \)
79 \( 1 - 11T + 79T^{2} \)
83 \( 1 + 13.8iT - 83T^{2} \)
89 \( 1 + (-6 + 3.46i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.5 + 2.59i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.44035778624950061147957872276, −9.125491814977131460741352980437, −8.616496818365215426066019536588, −8.021190000366835968610758794041, −6.57923219057040637207711233762, −5.83151206547036331881793263604, −4.81533594963346084835166617926, −3.94382124605847115562537405942, −1.97480537670654841789094656672, −0.862725209362693969245713652677, 1.82805174031164552467162764515, 3.43879295638917951737284003477, 4.05399802548037663462733162163, 5.49601565501557569533769550447, 6.45545279606363179680387294411, 7.15276901397403275853096472416, 8.177179616491761796300788596004, 9.306025942883328407102711541028, 10.23562244206659665065563228431, 10.83140782713892016440485690848

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