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Properties

Label 2-1216-152.125-c1-0-33
Degree $2$
Conductor $1216$
Sign $0.0443 + 0.999i$
Analytic cond. $9.70980$
Root an. cond. $3.11605$
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  + (1.80 + 1.04i)3-s + (−1.98 − 1.14i)5-s − 1.72·7-s + (0.675 + 1.17i)9-s − 2.75i·11-s + (−1.45 + 0.838i)13-s + (−2.39 − 4.13i)15-s + (0.0573 − 0.0994i)17-s + (3.55 − 2.52i)19-s + (−3.11 − 1.80i)21-s + (−2.20 − 3.82i)23-s + (0.125 + 0.217i)25-s − 3.43i·27-s + (3.68 − 2.12i)29-s − 1.36·31-s + ⋯
L(s)  = 1  + (1.04 + 0.602i)3-s + (−0.887 − 0.512i)5-s − 0.652·7-s + (0.225 + 0.390i)9-s − 0.830i·11-s + (−0.402 + 0.232i)13-s + (−0.617 − 1.06i)15-s + (0.0139 − 0.0241i)17-s + (0.815 − 0.578i)19-s + (−0.680 − 0.393i)21-s + (−0.460 − 0.797i)23-s + (0.0251 + 0.0435i)25-s − 0.661i·27-s + (0.685 − 0.395i)29-s − 0.244·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0443 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0443 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $0.0443 + 0.999i$
Analytic conductor: \(9.70980\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ 0.0443 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.266599875\)
\(L(\frac12)\) \(\approx\) \(1.266599875\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + (-3.55 + 2.52i)T \)
good3 \( 1 + (-1.80 - 1.04i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.98 + 1.14i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + 1.72T + 7T^{2} \)
11 \( 1 + 2.75iT - 11T^{2} \)
13 \( 1 + (1.45 - 0.838i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.0573 + 0.0994i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (2.20 + 3.82i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.68 + 2.12i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.36T + 31T^{2} \)
37 \( 1 + 4.06iT - 37T^{2} \)
41 \( 1 + (0.293 - 0.507i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.28 + 0.743i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.00 + 5.20i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-7.48 + 4.31i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (12.0 + 6.97i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-10.2 + 5.92i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.72 - 2.15i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.0279 + 0.0484i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.91 - 10.2i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.04 - 12.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 1.36iT - 83T^{2} \)
89 \( 1 + (-7.48 - 12.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.462 - 0.801i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.517885680013928015084555020876, −8.557821080386963540266737194033, −8.285047063141186651791078450039, −7.23736494517082403706189139638, −6.24651001436846195331821184175, −5.03182367796435036737108371755, −4.08031980505509219554849111644, −3.41528673610713591217773546936, −2.51313055263822309782497410086, −0.46430551673027719907483836134, 1.64968937955390942086896146878, 2.93043582589933133214886461842, 3.43369621377352198717657669384, 4.60600011612840681228987820185, 5.88459369284963189981217856162, 7.05869649850266665887140441952, 7.47844699292664121413283804496, 8.077081858724656024068922039466, 9.063941027646560244651104839281, 9.821640508583082348489022042664

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