Properties

Label 2-624-13.9-c1-0-11
Degree $2$
Conductor $624$
Sign $0.477 + 0.878i$
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 + 3·5-s + (−2 − 3.46i)7-s + (−0.499 − 0.866i)9-s + (2 − 3.46i)11-s + (−2.5 − 2.59i)13-s + (−1.5 + 2.59i)15-s + (−1.5 − 2.59i)17-s + (−2 − 3.46i)19-s + 3.99·21-s + (−4 + 6.92i)23-s + 4·25-s + 0.999·27-s + (2.5 − 4.33i)29-s + 8·31-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + 1.34·5-s + (−0.755 − 1.30i)7-s + (−0.166 − 0.288i)9-s + (0.603 − 1.04i)11-s + (−0.693 − 0.720i)13-s + (−0.387 + 0.670i)15-s + (−0.363 − 0.630i)17-s + (−0.458 − 0.794i)19-s + 0.872·21-s + (−0.834 + 1.44i)23-s + 0.800·25-s + 0.192·27-s + (0.464 − 0.804i)29-s + 1.43·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.477 + 0.878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.477 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $0.477 + 0.878i$
Analytic conductor: \(4.98266\)
Root analytic conductor: \(2.23218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :1/2),\ 0.477 + 0.878i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19246 - 0.709015i\)
\(L(\frac12)\) \(\approx\) \(1.19246 - 0.709015i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.5 + 2.59i)T \)
good5 \( 1 - 3T + 5T^{2} \)
7 \( 1 + (2 + 3.46i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2 + 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4 - 6.92i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.5 + 4.33i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + (3.5 - 6.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.5 + 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4 - 6.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 4T + 47T^{2} \)
53 \( 1 + 5T + 53T^{2} \)
59 \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.5 - 4.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4 + 6.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2 + 3.46i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 11T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7 - 12.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.26605442352060627644228041243, −9.769152958039800980759367859989, −9.082085633499943486008612595852, −7.75076916107353423681006090398, −6.59849027619078970003877556877, −6.03977718352932619505639831947, −4.96653572333780385766057812380, −3.81778754927339176247922855276, −2.67811407560731761810951929586, −0.78092302757712654319429896482, 1.91947071777838480588208054130, 2.46388135250376994776166240557, 4.36887885855977114378883232238, 5.54659780721599603499567235342, 6.36810786646446921750579328786, 6.74126309742186585395125833203, 8.290926747982199134972998442251, 9.180587457514239873106109126044, 9.801942763502910906073008206898, 10.54331968385547638521964653801

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