Properties

Label 2-624-39.32-c1-0-22
Degree $2$
Conductor $624$
Sign $-0.0431 + 0.999i$
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 + 0.402i)3-s + (−2.36 − 2.36i)5-s + (0.332 − 1.24i)7-s + (2.67 + 1.35i)9-s + (−1.52 − 5.69i)11-s + (−3.51 + 0.821i)13-s + (−3.03 − 4.93i)15-s + (1.04 + 1.80i)17-s + (−1.34 − 0.360i)19-s + (1.05 − 1.95i)21-s + (3.81 − 6.60i)23-s + 6.20i·25-s + (3.96 + 3.35i)27-s + (−6.87 − 3.96i)29-s + (−1.42 + 1.42i)31-s + ⋯
L(s)  = 1  + (0.972 + 0.232i)3-s + (−1.05 − 1.05i)5-s + (0.125 − 0.468i)7-s + (0.892 + 0.451i)9-s + (−0.460 − 1.71i)11-s + (−0.973 + 0.227i)13-s + (−0.783 − 1.27i)15-s + (0.252 + 0.436i)17-s + (−0.308 − 0.0827i)19-s + (0.230 − 0.426i)21-s + (0.795 − 1.37i)23-s + 1.24i·25-s + (0.762 + 0.646i)27-s + (−1.27 − 0.736i)29-s + (−0.256 + 0.256i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0431 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0431 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $-0.0431 + 0.999i$
Analytic conductor: \(4.98266\)
Root analytic conductor: \(2.23218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :1/2),\ -0.0431 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00725 - 1.05166i\)
\(L(\frac12)\) \(\approx\) \(1.00725 - 1.05166i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.68 - 0.402i)T \)
13 \( 1 + (3.51 - 0.821i)T \)
good5 \( 1 + (2.36 + 2.36i)T + 5iT^{2} \)
7 \( 1 + (-0.332 + 1.24i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (1.52 + 5.69i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-1.04 - 1.80i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.34 + 0.360i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-3.81 + 6.60i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (6.87 + 3.96i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.42 - 1.42i)T - 31iT^{2} \)
37 \( 1 + (-4.28 + 1.14i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-0.985 + 0.263i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-8.68 + 5.01i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.59 + 2.59i)T - 47iT^{2} \)
53 \( 1 - 7.08iT - 53T^{2} \)
59 \( 1 + (-1.48 - 0.397i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-3.39 - 5.88i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.22 - 8.30i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (1.21 - 4.53i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (9.61 + 9.61i)T + 73iT^{2} \)
79 \( 1 - 11.7T + 79T^{2} \)
83 \( 1 + (0.406 + 0.406i)T + 83iT^{2} \)
89 \( 1 + (-1.75 - 6.55i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (5.89 + 1.57i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.45151717345455024080329215470, −9.169031663785187958483470131338, −8.638758318660626508057596249144, −7.915431399575198665720536321132, −7.24353983545189853097641066097, −5.63657418172867891938655732777, −4.47798237176747108388259206119, −3.85714548292819651946775124851, −2.62386097459941416661421961183, −0.71301290512821209553825035797, 2.11738484088698072146901623559, 3.00737356985695312027246891242, 4.05636584586924601225541322261, 5.17105075795259500467171775316, 6.85306089948729827075664578278, 7.53860855888304309464873563396, 7.75701899870029053018372732180, 9.231246032385018307021160647774, 9.782174294626424128503442681936, 10.81108540368067784041523641968

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