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Properties

Label 2-624-39.20-c1-0-25
Degree $2$
Conductor $624$
Sign $-0.851 + 0.524i$
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  + (0.0795 − 1.73i)3-s + (2.02 − 2.02i)5-s + (−3.46 − 0.929i)7-s + (−2.98 − 0.275i)9-s + (4.05 − 1.08i)11-s + (−3.60 + 0.176i)13-s + (−3.33 − 3.65i)15-s + (−1.72 − 2.99i)17-s + (−0.581 + 2.16i)19-s + (−1.88 + 5.92i)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  + (0.0459 − 0.998i)3-s + (0.903 − 0.903i)5-s + (−1.31 − 0.351i)7-s + (−0.995 − 0.0917i)9-s + (1.22 − 0.327i)11-s + (−0.998 + 0.0490i)13-s + (−0.861 − 0.944i)15-s + (−0.418 − 0.725i)17-s + (−0.133 + 0.497i)19-s + (−0.410 + 1.29i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.851 + 0.524i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.851 + 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $-0.851 + 0.524i$
Analytic conductor: \(4.98266\)
Root analytic conductor: \(2.23218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :1/2),\ -0.851 + 0.524i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.340161 - 1.19974i\)
\(L(\frac12)\) \(\approx\) \(0.340161 - 1.19974i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0795 + 1.73i)T \)
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

−9.952991568986538963171637527226, −9.316032302908493336428937583125, −8.713802910501213203948955862126, −7.41003228320275479968880368993, −6.57363880881829225111554244489, −5.98696370660286078318488104239, −4.80155540591927632753308910181, −3.31939756336635833119781410537, −2.04046520252465164880483337398, −0.65602883869269338821104044489, 2.36163417082032982242917756800, 3.25171804962063882784883604298, 4.33924477498485398992327019350, 5.66494329213908232692017207640, 6.39205997562849553332122327417, 7.13309321698376274432150686640, 8.831877136988784173001267515880, 9.402243352891908832777555142161, 10.03344478497937690820268784712, 10.63450572089825942556926857152

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