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Properties

Label 2-1216-76.31-c1-0-0
Degree $2$
Conductor $1216$
Sign $-0.887 - 0.461i$
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  + (−0.982 − 1.70i)3-s + (0.349 + 0.605i)5-s + 3.80i·7-s + (−0.430 + 0.744i)9-s + 2.16i·11-s + (−1.16 − 0.672i)13-s + (0.686 − 1.18i)15-s + (−1.89 − 3.28i)17-s + (−1.62 + 4.04i)19-s + (6.46 − 3.73i)21-s + (−4.89 − 2.82i)23-s + (2.25 − 3.90i)25-s − 4.20·27-s + (−8.65 − 4.99i)29-s − 7.76·31-s + ⋯
L(s)  = 1  + (−0.567 − 0.982i)3-s + (0.156 + 0.270i)5-s + 1.43i·7-s + (−0.143 + 0.248i)9-s + 0.653i·11-s + (−0.323 − 0.186i)13-s + (0.177 − 0.307i)15-s + (−0.459 − 0.796i)17-s + (−0.372 + 0.928i)19-s + (1.41 − 0.814i)21-s + (−1.01 − 0.588i)23-s + (0.451 − 0.781i)25-s − 0.809·27-s + (−1.60 − 0.927i)29-s − 1.39·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.887 - 0.461i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.887 - 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $-0.887 - 0.461i$
Analytic conductor: \(9.70980\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (639, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ -0.887 - 0.461i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1052258449\)
\(L(\frac12)\) \(\approx\) \(0.1052258449\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.62 - 4.04i)T \)
good3 \( 1 + (0.982 + 1.70i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.349 - 0.605i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 - 3.80iT - 7T^{2} \)
11 \( 1 - 2.16iT - 11T^{2} \)
13 \( 1 + (1.16 + 0.672i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.89 + 3.28i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (4.89 + 2.82i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (8.65 + 4.99i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 7.76T + 31T^{2} \)
37 \( 1 - 1.31iT - 37T^{2} \)
41 \( 1 + (7.58 - 4.37i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.35 + 3.08i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.06 - 1.18i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.41 - 3.12i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.28 + 5.68i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.951 + 1.64i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.69 - 4.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.60 - 4.51i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-4.86 - 8.41i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.38 + 5.86i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.55iT - 83T^{2} \)
89 \( 1 + (1.43 + 0.827i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (9.10 - 5.25i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.967466987150960946165520191829, −9.308547908806763187052724063551, −8.346007894921600960307479326803, −7.49978211718722296178137038809, −6.69984287574709982469079500159, −5.93877142200240207183221045396, −5.34685217603021628380872250041, −4.04471717670539353682987546234, −2.48936096549231272333840592322, −1.87382408696336744318250762705, 0.04518693116869570399933857598, 1.73235003456435870438658007187, 3.60683015683843628922442457018, 4.09429918590598826620027157448, 5.07876236288811353795206020816, 5.78756404495192781481443000575, 6.97880678979906189343923337372, 7.60456931128306253457154587549, 8.825129557172126297379713103703, 9.474494470642116925182347578906

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