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Properties

Label 2-1216-152.107-c1-0-7
Degree $2$
Conductor $1216$
Sign $-0.304 - 0.952i$
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  + (0.121 − 0.0702i)3-s + (1.5 − 0.866i)5-s + 2i·7-s + (−1.49 + 2.58i)9-s + (−2.13 + 3.69i)13-s + (0.121 − 0.210i)15-s + (1.99 + 3.44i)17-s + (−4.31 + 0.630i)19-s + (0.140 + 0.243i)21-s + (−6.40 − 3.70i)23-s + (−1 + 1.73i)25-s + 0.839i·27-s + (1.99 − 3.44i)29-s − 8.62·31-s + (1.73 + 3i)35-s + ⋯
L(s)  = 1  + (0.0702 − 0.0405i)3-s + (0.670 − 0.387i)5-s + 0.755i·7-s + (−0.496 + 0.860i)9-s + (−0.590 + 1.02i)13-s + (0.0314 − 0.0543i)15-s + (0.482 + 0.836i)17-s + (−0.989 + 0.144i)19-s + (0.0306 + 0.0530i)21-s + (−1.33 − 0.771i)23-s + (−0.200 + 0.346i)25-s + 0.161i·27-s + (0.369 − 0.640i)29-s − 1.54·31-s + (0.292 + 0.507i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.304 - 0.952i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.304 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $-0.304 - 0.952i$
Analytic conductor: \(9.70980\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ -0.304 - 0.952i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.235653074\)
\(L(\frac12)\) \(\approx\) \(1.235653074\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.31 - 0.630i)T \)
good3 \( 1 + (-0.121 + 0.0702i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.5 + 0.866i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + (2.13 - 3.69i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.99 - 3.44i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (6.40 + 3.70i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.99 + 3.44i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 8.62T + 31T^{2} \)
37 \( 1 - 7.26T + 37T^{2} \)
41 \( 1 + (6.39 - 3.69i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.16 - 10.6i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.364 - 0.210i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.48 + 9.49i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.21 - 0.700i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.92 + 1.10i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.81 + 1.05i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.970 + 1.68i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.5 - 4.33i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.70 - 4.68i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 8.62T + 83T^{2} \)
89 \( 1 + (-14.5 - 8.39i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.39 - 3.69i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.851964274219244456106558781603, −9.180562398015760317858407738593, −8.363452894721871408843627462555, −7.74493215551036837809081104458, −6.40304955514082621228131980893, −5.83821929344037986243276536383, −4.95304807214290697016606340705, −3.99321090581671780475550468438, −2.41000041102627501412029117061, −1.87451264252166576248107898231, 0.48234309365699207345639393880, 2.16951257146972139665536456607, 3.23314402344832794208202516936, 4.14138259476655740867111786322, 5.45822027909683206502999475365, 6.04005234180005582847275140623, 7.08295975818626780033169200325, 7.70750281246000047428329364421, 8.800978555214558385541652872765, 9.574045001007696320701798444410

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