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Properties

Label 2-624-208.173-c1-0-10
Degree $2$
Conductor $624$
Sign $0.902 - 0.430i$
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  + (−0.634 − 1.26i)2-s + (0.965 − 0.258i)3-s + (−1.19 + 1.60i)4-s + (2.17 + 2.17i)5-s + (−0.939 − 1.05i)6-s + (0.763 + 1.32i)7-s + (2.78 + 0.494i)8-s + (0.866 − 0.499i)9-s + (1.36 − 4.12i)10-s + (−4.61 + 1.23i)11-s + (−0.740 + 1.85i)12-s + (0.0447 + 3.60i)13-s + (1.18 − 1.80i)14-s + (2.66 + 1.53i)15-s + (−1.14 − 3.83i)16-s + (2.06 + 3.57i)17-s + ⋯
L(s)  = 1  + (−0.448 − 0.893i)2-s + (0.557 − 0.149i)3-s + (−0.597 + 0.801i)4-s + (0.972 + 0.972i)5-s + (−0.383 − 0.431i)6-s + (0.288 + 0.499i)7-s + (0.984 + 0.174i)8-s + (0.288 − 0.166i)9-s + (0.433 − 1.30i)10-s + (−1.39 + 0.373i)11-s + (−0.213 + 0.536i)12-s + (0.0124 + 0.999i)13-s + (0.317 − 0.481i)14-s + (0.687 + 0.397i)15-s + (−0.285 − 0.958i)16-s + (0.499 + 0.865i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.902 - 0.430i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.902 - 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $0.902 - 0.430i$
Analytic conductor: \(4.98266\)
Root analytic conductor: \(2.23218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :1/2),\ 0.902 - 0.430i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40811 + 0.318802i\)
\(L(\frac12)\) \(\approx\) \(1.40811 + 0.318802i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.634 + 1.26i)T \)
3 \( 1 + (-0.965 + 0.258i)T \)
13 \( 1 + (-0.0447 - 3.60i)T \)
good5 \( 1 + (-2.17 - 2.17i)T + 5iT^{2} \)
7 \( 1 + (-0.763 - 1.32i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (4.61 - 1.23i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-2.06 - 3.57i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.11 + 1.36i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-3.24 - 1.87i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (8.48 - 2.27i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + 1.28iT - 31T^{2} \)
37 \( 1 + (-10.4 + 2.79i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (0.848 - 1.47i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.34 - 1.16i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 - 5.65iT - 47T^{2} \)
53 \( 1 + (-7.13 + 7.13i)T - 53iT^{2} \)
59 \( 1 + (-1.17 + 4.39i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-2.05 + 7.66i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (2.01 + 7.51i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (7.22 + 12.5i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 15.6T + 73T^{2} \)
79 \( 1 - 16.8T + 79T^{2} \)
83 \( 1 + (-2.38 + 2.38i)T - 83iT^{2} \)
89 \( 1 + (6.11 - 10.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.0798 - 0.0461i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.76666520742677605114567720247, −9.746981718985729755870440284555, −9.225290546240559394005049138623, −8.165920410404737233254115972883, −7.39434174326032529374372386287, −6.25698988895571541616443930535, −5.01378346474151321540434113102, −3.66670785617616440976032898543, −2.39987773191069446890349230638, −2.01572975619459764256949495529, 0.876527157361694565187323996373, 2.47572679383902886530374096308, 4.29001108398247591257992100935, 5.32590293094855144896108140156, 5.78000913179457514679481916534, 7.28154644955364227310431879716, 8.002889570910027233140671650969, 8.694243470372655525935676845168, 9.529886747235908100929765561301, 10.27208971290604684306806341604

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