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Properties

Label 2-624-208.173-c1-0-26
Degree $2$
Conductor $624$
Sign $0.729 - 0.684i$
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  + (0.140 + 1.40i)2-s + (0.965 − 0.258i)3-s + (−1.96 + 0.395i)4-s + (−1.36 − 1.36i)5-s + (0.499 + 1.32i)6-s + (0.523 + 0.906i)7-s + (−0.831 − 2.70i)8-s + (0.866 − 0.499i)9-s + (1.72 − 2.10i)10-s + (3.88 − 1.04i)11-s + (−1.79 + 0.889i)12-s + (3.52 − 0.758i)13-s + (−1.20 + 0.863i)14-s + (−1.66 − 0.963i)15-s + (3.68 − 1.54i)16-s + (0.669 + 1.15i)17-s + ⋯
L(s)  = 1  + (0.0993 + 0.995i)2-s + (0.557 − 0.149i)3-s + (−0.980 + 0.197i)4-s + (−0.609 − 0.609i)5-s + (0.204 + 0.540i)6-s + (0.197 + 0.342i)7-s + (−0.294 − 0.955i)8-s + (0.288 − 0.166i)9-s + (0.545 − 0.666i)10-s + (1.17 − 0.314i)11-s + (−0.517 + 0.256i)12-s + (0.977 − 0.210i)13-s + (−0.321 + 0.230i)14-s + (−0.430 − 0.248i)15-s + (0.921 − 0.387i)16-s + (0.162 + 0.281i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.729 - 0.684i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.729 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $0.729 - 0.684i$
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: \((2,\ 624,\ (\ :1/2),\ 0.729 - 0.684i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.59063 + 0.629486i\)
\(L(\frac12)\) \(\approx\) \(1.59063 + 0.629486i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.140 - 1.40i)T \)
3 \( 1 + (-0.965 + 0.258i)T \)
13 \( 1 + (-3.52 + 0.758i)T \)
good5 \( 1 + (1.36 + 1.36i)T + 5iT^{2} \)
7 \( 1 + (-0.523 - 0.906i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-3.88 + 1.04i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-0.669 - 1.15i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.17 - 0.316i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-0.473 - 0.273i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.08 + 0.557i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 - 10.6iT - 31T^{2} \)
37 \( 1 + (4.10 - 1.10i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-0.0933 + 0.161i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-9.28 - 2.48i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + 10.8iT - 47T^{2} \)
53 \( 1 + (0.909 - 0.909i)T - 53iT^{2} \)
59 \( 1 + (-3.02 + 11.2i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (2.69 - 10.0i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-0.427 - 1.59i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (6.61 + 11.4i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 5.26T + 73T^{2} \)
79 \( 1 + 7.40T + 79T^{2} \)
83 \( 1 + (-0.0260 + 0.0260i)T - 83iT^{2} \)
89 \( 1 + (3.83 - 6.64i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (14.2 - 8.19i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.58117353584123489898710264303, −9.383998861053203539214266395629, −8.516132603602451477571130459960, −8.410959810296964144894102646292, −7.17530566721608027348415586891, −6.33566936422980199018729826157, −5.28232440753147694026300138169, −4.16142608692723272262535222844, −3.39541039208075230642570949991, −1.16726475737502462509515438853, 1.30266725022128821296860779616, 2.76870803576670021131154649471, 3.84717912542651543130321411426, 4.31029993944625647496768894867, 5.86974239818323204448302417080, 7.13249529200766179262178561704, 8.022877295956377615948695023907, 9.048409802268104122207343474684, 9.604356069130678391236353712701, 10.71974879340874678494896316541

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