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Properties

Label 2-1216-152.107-c1-0-11
Degree $2$
Conductor $1216$
Sign $0.457 - 0.889i$
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  + (−1.5 + 0.866i)3-s + (2.36 − 1.36i)5-s − 1.26i·7-s − 4.46·11-s + (−2.36 + 4.09i)15-s + (2.73 + 4.73i)17-s + (−0.5 + 4.33i)19-s + (1.09 + 1.90i)21-s + (3.63 + 2.09i)23-s + (1.23 − 2.13i)25-s − 5.19i·27-s + (−1.09 + 1.90i)29-s + 2.19·31-s + (6.69 − 3.86i)33-s + (−1.73 − 3.00i)35-s + ⋯
L(s)  = 1  + (−0.866 + 0.499i)3-s + (1.05 − 0.610i)5-s − 0.479i·7-s − 1.34·11-s + (−0.610 + 1.05i)15-s + (0.662 + 1.14i)17-s + (−0.114 + 0.993i)19-s + (0.239 + 0.415i)21-s + (0.757 + 0.437i)23-s + (0.246 − 0.426i)25-s − 0.999i·27-s + (−0.203 + 0.353i)29-s + 0.394·31-s + (1.16 − 0.672i)33-s + (−0.292 − 0.507i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.457 - 0.889i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.457 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $0.457 - 0.889i$
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: \((2,\ 1216,\ (\ :1/2),\ 0.457 - 0.889i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.210529862\)
\(L(\frac12)\) \(\approx\) \(1.210529862\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 4.33i)T \)
good3 \( 1 + (1.5 - 0.866i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-2.36 + 1.36i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + 1.26iT - 7T^{2} \)
11 \( 1 + 4.46T + 11T^{2} \)
13 \( 1 + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.73 - 4.73i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-3.63 - 2.09i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.09 - 1.90i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.19T + 31T^{2} \)
37 \( 1 - 4.73T + 37T^{2} \)
41 \( 1 + (-6.69 + 3.86i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.73 - 3i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-8.36 - 4.83i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.73 - 8.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.696 + 0.401i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (10.5 + 6.09i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.69 - 2.13i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.26 - 2.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.5 - 4.33i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-8.19 - 14.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.39T + 83T^{2} \)
89 \( 1 + (11.1 + 6.46i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.69 - 3.86i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.01511318126344886763464365011, −9.311006178439811383837296458665, −8.184070942912604916848471730532, −7.53169879262876044552676143657, −6.07316529010924457819197622168, −5.67842533702992659940129787573, −4.98151030547790704149877157847, −4.00175221712662469712952313871, −2.54137437313929851049156270320, −1.19838153021153308554610381828, 0.64391011628248925473502425274, 2.35159060193111636819676012800, 2.94833798354960224839711385126, 4.83134756692990305500850222230, 5.53770292375002602066135710389, 6.14362082952280812590700884373, 6.97623755390816648034995337320, 7.71076479731846969601821714125, 8.957347656016976194096196700146, 9.637316344026697389116066371248

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