Properties

Label 2-1216-152.107-c1-0-15
Degree $2$
Conductor $1216$
Sign $0.923 + 0.382i$
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  + (−1.96 + 1.13i)3-s + (−1.5 + 0.866i)5-s − 2i·7-s + (1.08 − 1.87i)9-s + (−2.85 + 4.94i)13-s + (1.96 − 3.40i)15-s + (−0.583 − 1.01i)17-s + (0.144 − 4.35i)19-s + (2.27 + 3.93i)21-s + (−4.31 − 2.49i)23-s + (−1 + 1.73i)25-s − 1.89i·27-s + (0.583 − 1.01i)29-s − 0.289·31-s + (1.73 + 3i)35-s + ⋯
L(s)  = 1  + (−1.13 + 0.656i)3-s + (−0.670 + 0.387i)5-s − 0.755i·7-s + (0.361 − 0.625i)9-s + (−0.792 + 1.37i)13-s + (0.508 − 0.880i)15-s + (−0.141 − 0.245i)17-s + (0.0331 − 0.999i)19-s + (0.496 + 0.859i)21-s + (−0.900 − 0.519i)23-s + (−0.200 + 0.346i)25-s − 0.364i·27-s + (0.108 − 0.187i)29-s − 0.0519·31-s + (0.292 + 0.507i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $0.923 + 0.382i$
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: \((2,\ 1216,\ (\ :1/2),\ 0.923 + 0.382i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5970319750\)
\(L(\frac12)\) \(\approx\) \(0.5970319750\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.144 + 4.35i)T \)
good3 \( 1 + (1.96 - 1.13i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.5 - 0.866i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + 2iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + (2.85 - 4.94i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.583 + 1.01i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (4.31 + 2.49i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.583 + 1.01i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.289T + 31T^{2} \)
37 \( 1 - 2.71T + 37T^{2} \)
41 \( 1 + (-8.56 + 4.94i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.380 + 0.659i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.90 - 3.40i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.333 - 0.577i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-9.51 + 5.49i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.31 + 3.07i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-9.18 - 5.30i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.57 + 9.65i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.5 - 4.33i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.84 - 6.65i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 0.289T + 83T^{2} \)
89 \( 1 + (-14.0 - 8.12i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.56 + 4.94i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.805441823727724573886401125987, −9.126675433951560638951780062921, −7.83206807281547889934369671044, −7.08654984809847269850443078286, −6.42268804760546739115319324917, −5.32012439206509074044834381969, −4.38485639204841849152255168151, −3.98659421389969151756479164431, −2.40068158044340832002306850955, −0.42174003367055133225952337172, 0.837176119669689205186365979663, 2.34541586575194773938375335947, 3.71917255151559178142541590962, 4.90322281261633615062619330382, 5.73222135069048552430915219759, 6.17476005995548761109289431335, 7.47705676450186411633059994156, 7.902320335911857403722953818755, 8.846204073804086113508731615078, 9.956281850711730012485463361329

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