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Properties

Label 2-624-39.32-c1-0-25
Degree $2$
Conductor $624$
Sign $-0.766 - 0.642i$
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  + (−0.933 − 1.45i)3-s + (−2.76 − 2.76i)5-s + (0.657 − 2.45i)7-s + (−1.25 + 2.72i)9-s + (0.150 + 0.563i)11-s + (−1.20 − 3.39i)13-s + (−1.44 + 6.61i)15-s + (0.547 + 0.947i)17-s + (−1.32 − 0.355i)19-s + (−4.19 + 1.33i)21-s + (0.876 − 1.51i)23-s + 10.2i·25-s + (5.14 − 0.713i)27-s + (5.12 + 2.96i)29-s + (−6.49 + 6.49i)31-s + ⋯
L(s)  = 1  + (−0.539 − 0.842i)3-s + (−1.23 − 1.23i)5-s + (0.248 − 0.927i)7-s + (−0.418 + 0.908i)9-s + (0.0454 + 0.169i)11-s + (−0.335 − 0.942i)13-s + (−0.374 + 1.70i)15-s + (0.132 + 0.229i)17-s + (−0.304 − 0.0815i)19-s + (−0.915 + 0.290i)21-s + (0.182 − 0.316i)23-s + 2.05i·25-s + (0.990 − 0.137i)27-s + (0.952 + 0.549i)29-s + (−1.16 + 1.16i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $-0.766 - 0.642i$
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: \((2,\ 624,\ (\ :1/2),\ -0.766 - 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.153229 + 0.421041i\)
\(L(\frac12)\) \(\approx\) \(0.153229 + 0.421041i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.933 + 1.45i)T \)
13 \( 1 + (1.20 + 3.39i)T \)
good5 \( 1 + (2.76 + 2.76i)T + 5iT^{2} \)
7 \( 1 + (-0.657 + 2.45i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-0.150 - 0.563i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-0.547 - 0.947i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.32 + 0.355i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-0.876 + 1.51i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5.12 - 2.96i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (6.49 - 6.49i)T - 31iT^{2} \)
37 \( 1 + (2.98 - 0.801i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (5.11 - 1.37i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (3.26 - 1.88i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.51 + 5.51i)T - 47iT^{2} \)
53 \( 1 + 3.04iT - 53T^{2} \)
59 \( 1 + (8.19 + 2.19i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (4.67 + 8.09i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.70 + 6.37i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (0.220 - 0.821i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (5.18 + 5.18i)T + 73iT^{2} \)
79 \( 1 - 13.1T + 79T^{2} \)
83 \( 1 + (-5.15 - 5.15i)T + 83iT^{2} \)
89 \( 1 + (2.50 + 9.35i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (0.592 + 0.158i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.41165873175173398862689433245, −8.884597534860307853648597563212, −8.121357038592459280791363990118, −7.53883365669571080251205853708, −6.69933162074845742586188855870, −5.23202485734951436810304815944, −4.67960175933963378966901118215, −3.42987146210480942058893370027, −1.41648607087918997888036506502, −0.27790742184330532422488524846, 2.58454841647132047498691470690, 3.67904072074106581616203133993, 4.51398757027851571415388143964, 5.71624984723989489829127398419, 6.64846534298827173023534698651, 7.53722681467577362091702619688, 8.628118392602040941419866668529, 9.455959367428363541842877151906, 10.51338952872013730482106282625, 11.12922655950782221392496961086

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