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Properties

Label 2-1216-152.45-c1-0-12
Degree $2$
Conductor $1216$
Sign $0.678 - 0.734i$
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  + (−2.17 + 1.25i)3-s + (0.891 − 0.514i)5-s − 4.18·7-s + (1.66 − 2.87i)9-s − 2.06i·11-s + (2.25 + 1.29i)13-s + (−1.29 + 2.24i)15-s + (−3.49 − 6.04i)17-s + (1.09 + 4.22i)19-s + (9.10 − 5.25i)21-s + (2.14 − 3.71i)23-s + (−1.96 + 3.41i)25-s + 0.806i·27-s + (5.17 + 2.98i)29-s + 2.70·31-s + ⋯
L(s)  = 1  + (−1.25 + 0.725i)3-s + (0.398 − 0.230i)5-s − 1.58·7-s + (0.553 − 0.958i)9-s − 0.621i·11-s + (0.624 + 0.360i)13-s + (−0.334 + 0.578i)15-s + (−0.846 − 1.46i)17-s + (0.250 + 0.968i)19-s + (1.98 − 1.14i)21-s + (0.447 − 0.775i)23-s + (−0.393 + 0.682i)25-s + 0.155i·27-s + (0.960 + 0.554i)29-s + 0.485·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $0.678 - 0.734i$
Analytic conductor: \(9.70980\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (1185, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ 0.678 - 0.734i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7670627313\)
\(L(\frac12)\) \(\approx\) \(0.7670627313\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.09 - 4.22i)T \)
good3 \( 1 + (2.17 - 1.25i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.891 + 0.514i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + 4.18T + 7T^{2} \)
11 \( 1 + 2.06iT - 11T^{2} \)
13 \( 1 + (-2.25 - 1.29i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.49 + 6.04i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-2.14 + 3.71i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5.17 - 2.98i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.70T + 31T^{2} \)
37 \( 1 - 9.93iT - 37T^{2} \)
41 \( 1 + (2.42 + 4.20i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-8.57 + 4.94i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.86 - 4.96i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.00 + 2.31i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.382 - 0.220i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.59 - 2.07i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.07 - 0.618i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-7.69 - 13.3i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.846 - 1.46i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.13 - 10.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 8.92iT - 83T^{2} \)
89 \( 1 + (-5.55 + 9.62i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.11 - 7.12i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.853304708741830630764726832871, −9.342231394835701556256329518925, −8.456487899005276026924056429479, −6.89089479426972615425275185094, −6.41083673339329234262650903432, −5.66397701605444307072405415685, −4.86633463958556416928242362347, −3.82251964977739076173575060881, −2.80943141044063938207125796411, −0.77359649722688091005104462971, 0.59635754712511935744954954597, 2.10916785560634191491837839767, 3.37887900630123898118702716210, 4.59865973855387011987784202111, 5.83263389071382885314674442862, 6.30962563179302913959431013546, 6.75461274896438586539351016689, 7.73063897049410508197108021765, 8.964554297018654536133366955935, 9.753310850658177894083213343049

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