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Properties

Label 2-624-52.11-c1-0-9
Degree $2$
Conductor $624$
Sign $-0.120 + 0.992i$
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  + (−0.866 + 0.5i)3-s + (−0.0648 − 0.0648i)5-s + (−2.75 + 0.739i)7-s + (0.499 − 0.866i)9-s + (0.150 − 0.561i)11-s + (0.572 − 3.55i)13-s + (0.0886 + 0.0237i)15-s + (−0.0836 − 0.0482i)17-s + (0.134 + 0.502i)19-s + (2.01 − 2.01i)21-s + (−4.23 − 7.33i)23-s − 4.99i·25-s + 0.999i·27-s + (−1.16 − 2.02i)29-s + (1.90 − 1.90i)31-s + ⋯
L(s)  = 1  + (−0.499 + 0.288i)3-s + (−0.0290 − 0.0290i)5-s + (−1.04 + 0.279i)7-s + (0.166 − 0.288i)9-s + (0.0453 − 0.169i)11-s + (0.158 − 0.987i)13-s + (0.0228 + 0.00613i)15-s + (−0.0202 − 0.0117i)17-s + (0.0309 + 0.115i)19-s + (0.440 − 0.440i)21-s + (−0.883 − 1.52i)23-s − 0.998i·25-s + 0.192i·27-s + (−0.216 − 0.375i)29-s + (0.342 − 0.342i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.120 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.120 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $-0.120 + 0.992i$
Analytic conductor: \(4.98266\)
Root analytic conductor: \(2.23218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :1/2),\ -0.120 + 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.442028 - 0.498888i\)
\(L(\frac12)\) \(\approx\) \(0.442028 - 0.498888i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (-0.572 + 3.55i)T \)
good5 \( 1 + (0.0648 + 0.0648i)T + 5iT^{2} \)
7 \( 1 + (2.75 - 0.739i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-0.150 + 0.561i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (0.0836 + 0.0482i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.134 - 0.502i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (4.23 + 7.33i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.16 + 2.02i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.90 + 1.90i)T - 31iT^{2} \)
37 \( 1 + (-5.81 - 1.55i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-2.28 + 8.53i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (2.94 - 5.10i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.76 + 5.76i)T + 47iT^{2} \)
53 \( 1 + 9.07T + 53T^{2} \)
59 \( 1 + (-3.62 + 0.970i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (1.53 - 2.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.17 - 1.38i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (2.94 + 10.9i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-0.619 + 0.619i)T - 73iT^{2} \)
79 \( 1 - 1.24iT - 79T^{2} \)
83 \( 1 + (-5.92 + 5.92i)T - 83iT^{2} \)
89 \( 1 + (2.34 + 0.628i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (9.69 - 2.59i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.24318824724949945630786209199, −9.770582394181458904762579823173, −8.655691571053523239312051414355, −7.80691883612999410487669139244, −6.40990635770257230767355705326, −6.06491792289985262791708089460, −4.82947626219832459161310743351, −3.71345359091534145318825207935, −2.57215664680655656287136831796, −0.38960454570243438390625551263, 1.57667670364326387787088752244, 3.21011115673605442493300515687, 4.27394267192905720876786577537, 5.52049261355101576051883984665, 6.45284276182530138929295681927, 7.10790209636117164815321315847, 8.086785003371940619031481423697, 9.467721990662567157754518570172, 9.737996845296130704323670108649, 11.02988269987092663022914613377

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