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Properties

Label 2-624-13.4-c1-0-8
Degree $2$
Conductor $624$
Sign $0.702 + 0.711i$
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 − 0.267i·5-s + (−0.633 + 0.366i)7-s + (−0.499 − 0.866i)9-s + (4.09 + 2.36i)11-s + (2.59 − 2.5i)13-s + (−0.232 − 0.133i)15-s + (−1.13 − 1.96i)17-s + (1.09 − 0.633i)19-s + 0.732i·21-s + (3.09 − 5.36i)23-s + 4.92·25-s − 0.999·27-s + (−1.23 + 2.13i)29-s − 5.46i·31-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s − 0.119i·5-s + (−0.239 + 0.138i)7-s + (−0.166 − 0.288i)9-s + (1.23 + 0.713i)11-s + (0.720 − 0.693i)13-s + (−0.0599 − 0.0345i)15-s + (−0.275 − 0.476i)17-s + (0.251 − 0.145i)19-s + 0.159i·21-s + (0.645 − 1.11i)23-s + 0.985·25-s − 0.192·27-s + (−0.228 + 0.396i)29-s − 0.981i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.702 + 0.711i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.702 + 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $0.702 + 0.711i$
Analytic conductor: \(4.98266\)
Root analytic conductor: \(2.23218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :1/2),\ 0.702 + 0.711i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.57933 - 0.660123i\)
\(L(\frac12)\) \(\approx\) \(1.57933 - 0.660123i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.59 + 2.5i)T \)
good5 \( 1 + 0.267iT - 5T^{2} \)
7 \( 1 + (0.633 - 0.366i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-4.09 - 2.36i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.13 + 1.96i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.09 + 0.633i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.09 + 5.36i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.23 - 2.13i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 5.46iT - 31T^{2} \)
37 \( 1 + (9.06 + 5.23i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-9.86 - 5.69i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.83 + 6.63i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 8.19iT - 47T^{2} \)
53 \( 1 - 0.464T + 53T^{2} \)
59 \( 1 + (6.92 - 4i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.598 + 1.03i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-9.63 - 5.56i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.09 + 0.633i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 9.73iT - 73T^{2} \)
79 \( 1 - 9.46T + 79T^{2} \)
83 \( 1 - 10.1iT - 83T^{2} \)
89 \( 1 + (-2.19 - 1.26i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.19 + 3i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.57677794781225045111296256171, −9.310122639957589206313806802661, −8.945581041985939672453921832613, −7.82420297013358269313464485386, −6.89140201592224606500353821929, −6.20525699269522455494798904006, −4.93175656160080098424383277802, −3.76807867685129930061042082637, −2.57995845748586323648566589504, −1.09935040662624855569028054693, 1.49182821399345518166288024057, 3.27748738744588351168379374794, 3.90299633300606398718795154443, 5.15462137014560482086804505998, 6.32404254896505002427804470460, 7.01692139927745341119979734936, 8.391105197685961634187259587762, 8.977812883619283936323426681445, 9.727838671998073031428649936489, 10.81567610084212723285475585461

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