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Properties

Label 2-1216-152.27-c1-0-38
Degree $2$
Conductor $1216$
Sign $0.0489 - 0.998i$
Analytic cond. $9.70980$
Root an. cond. $3.11605$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0489 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0489 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $0.0489 - 0.998i$
Analytic conductor: \(9.70980\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (863, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ 0.0489 - 0.998i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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

−9.054594348370106354959594357225, −8.052420101715040672955909364341, −7.47764733621339253762428209427, −6.77523702346880101930218217442, −5.53950627704779727473066817125, −4.99830295761809621610591852870, −3.95040392874279238336211538598, −2.76398728565106356709405996052, −0.922295206936550637962057402849, 0, 2.25620591739892058766358954931, 3.49486431902112893523638122725, 4.31253683958934029031937233432, 5.47472899284056516914550446967, 5.88371764144996900914480826913, 7.09846534150064078162454146164, 8.064341166346484411623735593355, 8.362119013415235139609880166602, 9.975004536878591871849470234078

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