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Properties

Label 2-624-39.2-c1-0-24
Degree $2$
Conductor $624$
Sign $-0.919 + 0.391i$
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  + (1.45 + 0.933i)3-s + (−2.02 − 2.02i)5-s + (−3.46 + 0.929i)7-s + (1.25 + 2.72i)9-s + (−4.05 − 1.08i)11-s + (−3.60 − 0.176i)13-s + (−1.06 − 4.83i)15-s + (1.72 − 2.99i)17-s + (−0.581 − 2.16i)19-s + (−5.92 − 1.88i)21-s + (−1.51 − 2.62i)23-s + 3.16i·25-s + (−0.713 + 5.14i)27-s + (−1.74 + 1.00i)29-s + (−1.21 + 1.21i)31-s + ⋯
L(s)  = 1  + (0.842 + 0.539i)3-s + (−0.903 − 0.903i)5-s + (−1.31 + 0.351i)7-s + (0.418 + 0.908i)9-s + (−1.22 − 0.327i)11-s + (−0.998 − 0.0490i)13-s + (−0.273 − 1.24i)15-s + (0.418 − 0.725i)17-s + (−0.133 − 0.497i)19-s + (−1.29 − 0.410i)21-s + (−0.316 − 0.547i)23-s + 0.633i·25-s + (−0.137 + 0.990i)27-s + (−0.323 + 0.187i)29-s + (−0.218 + 0.218i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.919 + 0.391i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.919 + 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $-0.919 + 0.391i$
Analytic conductor: \(4.98266\)
Root analytic conductor: \(2.23218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :1/2),\ -0.919 + 0.391i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0361835 - 0.177240i\)
\(L(\frac12)\) \(\approx\) \(0.0361835 - 0.177240i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.45 - 0.933i)T \)
13 \( 1 + (3.60 + 0.176i)T \)
good5 \( 1 + (2.02 + 2.02i)T + 5iT^{2} \)
7 \( 1 + (3.46 - 0.929i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (4.05 + 1.08i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-1.72 + 2.99i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.581 + 2.16i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (1.51 + 2.62i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.74 - 1.00i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.21 - 1.21i)T - 31iT^{2} \)
37 \( 1 + (1.25 - 4.67i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (2.05 - 7.67i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-1.68 - 0.975i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.957 - 0.957i)T - 47iT^{2} \)
53 \( 1 + 7.22iT - 53T^{2} \)
59 \( 1 + (2.66 + 9.93i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (0.137 - 0.237i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.10 + 1.09i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-10.6 + 2.85i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-10.0 - 10.0i)T + 73iT^{2} \)
79 \( 1 + 1.58T + 79T^{2} \)
83 \( 1 + (2.58 + 2.58i)T + 83iT^{2} \)
89 \( 1 + (9.50 + 2.54i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (2.07 + 7.74i)T + (-84.0 + 48.5i)T^{2} \)
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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

−9.889828113315816071382311057432, −9.474270813997509700703374915795, −8.420846230781593653991368175898, −7.889467609881821989718907732104, −6.84550536585008919093857212669, −5.30388615056262842228348083782, −4.62072078785348378097886209547, −3.36771438393604083823988120909, −2.60520104271857448812968848213, −0.081847277570407172359488960339, 2.33483926608813321203485940911, 3.28643005818126058443275531784, 3.99917361315855544992213996154, 5.75256433812847688239582003887, 6.92049446312250112647181779049, 7.44227731671400349699569411049, 8.050031815738629948273318152539, 9.343125628950254542327765592444, 10.10610163581781686505469299358, 10.76401643696709312364127264283

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