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Properties

Label 2-1216-152.107-c1-0-23
Degree $2$
Conductor $1216$
Sign $0.967 + 0.251i$
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  + (0.132 − 0.0766i)3-s + (−1.14 + 0.658i)5-s − 1.40i·7-s + (−1.48 + 2.57i)9-s + 3.27·11-s + (2.31 − 4.01i)13-s + (−0.100 + 0.174i)15-s + (−2.24 − 3.89i)17-s + (2.56 + 3.52i)19-s + (−0.107 − 0.186i)21-s + (−0.100 − 0.0582i)23-s + (−1.63 + 2.82i)25-s + 0.915i·27-s + (3.34 − 5.78i)29-s − 2.43·31-s + ⋯
L(s)  = 1  + (0.0766 − 0.0442i)3-s + (−0.509 + 0.294i)5-s − 0.531i·7-s + (−0.496 + 0.859i)9-s + 0.987·11-s + (0.642 − 1.11i)13-s + (−0.0260 + 0.0451i)15-s + (−0.545 − 0.945i)17-s + (0.588 + 0.808i)19-s + (−0.0235 − 0.0407i)21-s + (−0.0210 − 0.0121i)23-s + (−0.326 + 0.565i)25-s + 0.176i·27-s + (0.620 − 1.07i)29-s − 0.437·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.251i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 + 0.251i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $0.967 + 0.251i$
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: \((2,\ 1216,\ (\ :1/2),\ 0.967 + 0.251i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.568926759\)
\(L(\frac12)\) \(\approx\) \(1.568926759\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.56 - 3.52i)T \)
good3 \( 1 + (-0.132 + 0.0766i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.14 - 0.658i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + 1.40iT - 7T^{2} \)
11 \( 1 - 3.27T + 11T^{2} \)
13 \( 1 + (-2.31 + 4.01i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.24 + 3.89i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (0.100 + 0.0582i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.34 + 5.78i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 2.43T + 31T^{2} \)
37 \( 1 - 8.87T + 37T^{2} \)
41 \( 1 + (-5.96 + 3.44i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.01 - 6.95i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-7.69 - 4.44i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.47 + 4.27i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-9.19 + 5.30i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (9.55 + 5.51i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.900 - 0.520i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.84 - 4.93i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.104 - 0.181i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.20 + 14.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 16.9T + 83T^{2} \)
89 \( 1 + (5.18 + 2.99i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.71 - 3.30i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.693895400319244180519223275352, −8.867052245413018772176123973007, −7.74410127953494944124252633894, −7.60724363998070136148661072703, −6.33431454535853055808688270115, −5.55269117746240993152167733906, −4.38885278837186629607645508152, −3.55570637486743332689920746609, −2.50835512156014494215808008993, −0.887587720908602762068146889183, 1.05487210618224610067294906197, 2.51907619551824507649920468514, 3.83803748813374007609788237452, 4.30734986486148608190269423085, 5.71568718147624167378350697149, 6.41126470233683391354279468828, 7.19088322334562793817443736867, 8.448715908833415306403288856184, 8.967044407291650318103157972911, 9.381442055688511434962018004277

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