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Properties

Label 2-624-39.2-c1-0-0
Degree $2$
Conductor $624$
Sign $0.242 - 0.970i$
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  + (−1.55 − 0.756i)3-s + (−3.14 − 3.14i)5-s + (−2.26 + 0.607i)7-s + (1.85 + 2.35i)9-s + (−3.54 − 0.950i)11-s + (2.62 − 2.47i)13-s + (2.52 + 7.28i)15-s + (0.814 − 1.41i)17-s + (1.43 + 5.36i)19-s + (3.98 + 0.768i)21-s + (0.495 + 0.858i)23-s + 14.8i·25-s + (−1.10 − 5.07i)27-s + (−3.01 + 1.74i)29-s + (4.72 − 4.72i)31-s + ⋯
L(s)  = 1  + (−0.899 − 0.436i)3-s + (−1.40 − 1.40i)5-s + (−0.856 + 0.229i)7-s + (0.618 + 0.785i)9-s + (−1.07 − 0.286i)11-s + (0.726 − 0.686i)13-s + (0.651 + 1.88i)15-s + (0.197 − 0.342i)17-s + (0.329 + 1.23i)19-s + (0.870 + 0.167i)21-s + (0.103 + 0.178i)23-s + 2.96i·25-s + (−0.213 − 0.976i)27-s + (−0.560 + 0.323i)29-s + (0.848 − 0.848i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.242 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.242 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $0.242 - 0.970i$
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: \((2,\ 624,\ (\ :1/2),\ 0.242 - 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.142823 + 0.111568i\)
\(L(\frac12)\) \(\approx\) \(0.142823 + 0.111568i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.55 + 0.756i)T \)
13 \( 1 + (-2.62 + 2.47i)T \)
good5 \( 1 + (3.14 + 3.14i)T + 5iT^{2} \)
7 \( 1 + (2.26 - 0.607i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (3.54 + 0.950i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-0.814 + 1.41i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.43 - 5.36i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-0.495 - 0.858i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.01 - 1.74i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.72 + 4.72i)T - 31iT^{2} \)
37 \( 1 + (-0.485 + 1.81i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (1.42 - 5.31i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-2.75 - 1.59i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.43 + 1.43i)T - 47iT^{2} \)
53 \( 1 - 9.03iT - 53T^{2} \)
59 \( 1 + (0.206 + 0.771i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (0.868 - 1.50i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (12.5 + 3.37i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (4.81 - 1.28i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (5.36 + 5.36i)T + 73iT^{2} \)
79 \( 1 + 4.28T + 79T^{2} \)
83 \( 1 + (-5.71 - 5.71i)T + 83iT^{2} \)
89 \( 1 + (5.12 + 1.37i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-1.80 - 6.73i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.99186186072790521251281766527, −10.02716485796720002354284789018, −8.913600599282214111458287481595, −7.87665561262369603985433744080, −7.60968994892337950241090727491, −6.04547133931004197749607324266, −5.39953336477268367411370719261, −4.39282447610356183408809413125, −3.24950076936377968846133439933, −1.07093309795482837634868732302, 0.13691296506027245949830759476, 2.91123093107302851445760970771, 3.76083351593890126547925740971, 4.69072270021192015960404754690, 6.10610457513219285419132929805, 6.88875986642483206240518701855, 7.42194194711987966479583191519, 8.670641274545424866370146309779, 9.979291101473279671339498357254, 10.53348590101771585207625869122

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