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Properties

Label 2-624-13.8-c2-0-13
Degree $2$
Conductor $624$
Sign $0.839 - 0.543i$
Analytic cond. $17.0027$
Root an. cond. $4.12344$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73·3-s + (4.73 + 4.73i)5-s + (2.73 − 2.73i)7-s + 2.99·9-s + (−1.73 + 1.73i)11-s + (9.92 + 8.39i)13-s + (8.19 + 8.19i)15-s − 29.3i·17-s + (11.2 + 11.2i)19-s + (4.73 − 4.73i)21-s + 29.3i·23-s + 19.7i·25-s + 5.19·27-s + 31.8·29-s + (−26.9 − 26.9i)31-s + ⋯
L(s)  = 1  + 0.577·3-s + (0.946 + 0.946i)5-s + (0.390 − 0.390i)7-s + 0.333·9-s + (−0.157 + 0.157i)11-s + (0.763 + 0.645i)13-s + (0.546 + 0.546i)15-s − 1.72i·17-s + (0.593 + 0.593i)19-s + (0.225 − 0.225i)21-s + 1.27i·23-s + 0.791i·25-s + 0.192·27-s + 1.09·29-s + (−0.870 − 0.870i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.839 - 0.543i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.839 - 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $0.839 - 0.543i$
Analytic conductor: \(17.0027\)
Root analytic conductor: \(4.12344\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (385, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :1),\ 0.839 - 0.543i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.859361862\)
\(L(\frac12)\) \(\approx\) \(2.859361862\)
\(L(2)\) 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 \( 1 - 1.73T \)
13 \( 1 + (-9.92 - 8.39i)T \)
good5 \( 1 + (-4.73 - 4.73i)T + 25iT^{2} \)
7 \( 1 + (-2.73 + 2.73i)T - 49iT^{2} \)
11 \( 1 + (1.73 - 1.73i)T - 121iT^{2} \)
17 \( 1 + 29.3iT - 289T^{2} \)
19 \( 1 + (-11.2 - 11.2i)T + 361iT^{2} \)
23 \( 1 - 29.3iT - 529T^{2} \)
29 \( 1 - 31.8T + 841T^{2} \)
31 \( 1 + (26.9 + 26.9i)T + 961iT^{2} \)
37 \( 1 + (30.8 - 30.8i)T - 1.36e3iT^{2} \)
41 \( 1 + (14.4 + 14.4i)T + 1.68e3iT^{2} \)
43 \( 1 - 25.1iT - 1.84e3T^{2} \)
47 \( 1 + (-41.1 + 41.1i)T - 2.20e3iT^{2} \)
53 \( 1 - 2.28T + 2.80e3T^{2} \)
59 \( 1 + (54.6 - 54.6i)T - 3.48e3iT^{2} \)
61 \( 1 + 7.42T + 3.72e3T^{2} \)
67 \( 1 + (-60.6 - 60.6i)T + 4.48e3iT^{2} \)
71 \( 1 + (38.9 + 38.9i)T + 5.04e3iT^{2} \)
73 \( 1 + (-40.3 + 40.3i)T - 5.32e3iT^{2} \)
79 \( 1 - 148.T + 6.24e3T^{2} \)
83 \( 1 + (73.7 + 73.7i)T + 6.88e3iT^{2} \)
89 \( 1 + (-25.5 + 25.5i)T - 7.92e3iT^{2} \)
97 \( 1 + (86.0 + 86.0i)T + 9.40e3iT^{2} \)
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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

−10.33073406584841124688900617879, −9.661661296224919847749954737868, −8.919422717029617790275304020580, −7.65821964047608008079358953825, −7.05629380511245560899401631512, −6.05429798933141812804919540352, −4.96993042703925473549228725476, −3.63715817110066708970858429330, −2.63325534949399127132802165630, −1.47696987016733656220328235886, 1.16802306390174666145178728290, 2.23560703719162069751026186857, 3.60130345807705741553001921617, 4.88995183867008889795561638377, 5.66691823736142500484365231229, 6.62335156687739434585996696026, 8.093775964968884539719770658857, 8.596341821686895814506943152117, 9.229043767332572027176409659220, 10.33199025091810847242397098859

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