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Properties

Label 2-624-208.173-c1-0-11
Degree $2$
Conductor $624$
Sign $0.666 - 0.745i$
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  + (−1.23 + 0.697i)2-s + (−0.965 + 0.258i)3-s + (1.02 − 1.71i)4-s + (0.334 + 0.334i)5-s + (1.00 − 0.992i)6-s + (0.820 + 1.42i)7-s + (−0.0665 + 2.82i)8-s + (0.866 − 0.499i)9-s + (−0.644 − 0.177i)10-s + (2.84 − 0.763i)11-s + (−0.547 + 1.92i)12-s + (−0.734 − 3.52i)13-s + (−1.99 − 1.17i)14-s + (−0.409 − 0.236i)15-s + (−1.89 − 3.52i)16-s + (1.61 + 2.79i)17-s + ⋯
L(s)  = 1  + (−0.869 + 0.493i)2-s + (−0.557 + 0.149i)3-s + (0.513 − 0.858i)4-s + (0.149 + 0.149i)5-s + (0.411 − 0.405i)6-s + (0.309 + 0.536i)7-s + (−0.0235 + 0.999i)8-s + (0.288 − 0.166i)9-s + (−0.203 − 0.0562i)10-s + (0.858 − 0.230i)11-s + (−0.158 + 0.555i)12-s + (−0.203 − 0.979i)13-s + (−0.534 − 0.314i)14-s + (−0.105 − 0.0609i)15-s + (−0.472 − 0.881i)16-s + (0.391 + 0.677i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.666 - 0.745i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.666 - 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $0.666 - 0.745i$
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: \((2,\ 624,\ (\ :1/2),\ 0.666 - 0.745i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.825960 + 0.369456i\)
\(L(\frac12)\) \(\approx\) \(0.825960 + 0.369456i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.23 - 0.697i)T \)
3 \( 1 + (0.965 - 0.258i)T \)
13 \( 1 + (0.734 + 3.52i)T \)
good5 \( 1 + (-0.334 - 0.334i)T + 5iT^{2} \)
7 \( 1 + (-0.820 - 1.42i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.84 + 0.763i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-1.61 - 2.79i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.05 + 1.08i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-4.63 - 2.67i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.54 + 0.413i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 - 2.93iT - 31T^{2} \)
37 \( 1 + (-8.35 + 2.23i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (2.57 - 4.46i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-12.2 - 3.29i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 - 5.43iT - 47T^{2} \)
53 \( 1 + (-1.92 + 1.92i)T - 53iT^{2} \)
59 \( 1 + (2.75 - 10.2i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (1.19 - 4.46i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-2.06 - 7.70i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-6.79 - 11.7i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 2.99T + 73T^{2} \)
79 \( 1 + 3.61T + 79T^{2} \)
83 \( 1 + (-9.94 + 9.94i)T - 83iT^{2} \)
89 \( 1 + (-0.745 + 1.29i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.68 - 2.12i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.64280013276320426512213220077, −9.848684242906483544869414308488, −8.933567785260033247187486652746, −8.204341196700226505490299029027, −7.18480948827033659581454695143, −6.17044042584261009703076997745, −5.64483751343675532637140070755, −4.42887842168580222127701555250, −2.66011990413289878882619945402, −1.08488687541724951837638412327, 0.950612487105421752932004533615, 2.18065948334119216226983547397, 3.82160289819926332702368377644, 4.76387988416971260595216027125, 6.31194149826768655409231743285, 7.03597031537890086447962826757, 7.83653198088524697510189183995, 9.064947995250596094106896013642, 9.511603008017227666781196231546, 10.64593779179927355689505404245

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