Properties

Label 2-624-39.2-c1-0-1
Degree $2$
Conductor $624$
Sign $-0.766 - 0.642i$
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  + (−1.64 + 0.529i)3-s + (−1.69 − 1.69i)5-s + (1.36 − 0.366i)7-s + (2.43 − 1.74i)9-s + (−1.69 − 0.453i)11-s + (−1.59 + 3.23i)13-s + (3.68 + 1.89i)15-s + (−1.07 + 1.85i)17-s + (0.267 + i)19-s + (−2.05 + 1.32i)21-s + 0.732i·25-s + (−3.09 + 4.17i)27-s + (−4.79 + 2.76i)29-s + (−4.46 + 4.46i)31-s + (3.03 − 0.148i)33-s + ⋯
L(s)  = 1  + (−0.952 + 0.305i)3-s + (−0.757 − 0.757i)5-s + (0.516 − 0.138i)7-s + (0.813 − 0.582i)9-s + (−0.510 − 0.136i)11-s + (−0.443 + 0.896i)13-s + (0.952 + 0.489i)15-s + (−0.260 + 0.450i)17-s + (0.0614 + 0.229i)19-s + (−0.449 + 0.289i)21-s + 0.146i·25-s + (−0.596 + 0.802i)27-s + (−0.889 + 0.513i)29-s + (−0.801 + 0.801i)31-s + (0.527 − 0.0258i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $-0.766 - 0.642i$
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: \((2,\ 624,\ (\ :1/2),\ -0.766 - 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0915997 + 0.251752i\)
\(L(\frac12)\) \(\approx\) \(0.0915997 + 0.251752i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.64 - 0.529i)T \)
13 \( 1 + (1.59 - 3.23i)T \)
good5 \( 1 + (1.69 + 1.69i)T + 5iT^{2} \)
7 \( 1 + (-1.36 + 0.366i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (1.69 + 0.453i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.07 - 1.85i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.267 - i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.79 - 2.76i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.46 - 4.46i)T - 31iT^{2} \)
37 \( 1 + (1.76 - 6.59i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-0.166 + 0.619i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (7.09 + 4.09i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.77 - 6.77i)T - 47iT^{2} \)
53 \( 1 - 4.62iT - 53T^{2} \)
59 \( 1 + (-1.23 - 4.62i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.46 - 2.26i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-4.62 + 1.23i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (6.09 + 6.09i)T + 73iT^{2} \)
79 \( 1 + 2T + 79T^{2} \)
83 \( 1 + (1.23 + 1.23i)T + 83iT^{2} \)
89 \( 1 + (-9.70 - 2.60i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-3.36 - 12.5i)T + (-84.0 + 48.5i)T^{2} \)
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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

−11.08633214658681090980315866124, −10.23071442746105988014194499287, −9.228156967753853324688796547714, −8.325707466322068178110137109567, −7.40225645595082475349536913802, −6.43823393895604175316492904299, −5.19404842280901604852814749791, −4.65045796969550341235098523409, −3.66033525275092640227061942354, −1.56227927009151504941502874209, 0.16621611118622402308105114894, 2.17633209303968847251292465081, 3.60481636612753592979344563431, 4.89726205800900926835667728752, 5.61145373343759312201609727240, 6.84205995144051550526833597428, 7.50712160272790781285297382236, 8.172546532785626399053270052150, 9.664316123652437107477802232083, 10.49070525788965713712693684782

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