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Properties

Label 2-1148-41.31-c1-0-1
Degree $2$
Conductor $1148$
Sign $-0.960 + 0.279i$
Analytic cond. $9.16682$
Root an. cond. $3.02767$
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  + 1.06i·3-s + (0.727 + 2.24i)5-s + (−0.587 − 0.809i)7-s + 1.85·9-s + (−5.06 − 1.64i)11-s + (−3.77 + 5.19i)13-s + (−2.39 + 0.778i)15-s + (−6.88 − 2.23i)17-s + (−2.95 − 4.06i)19-s + (0.865 − 0.628i)21-s + (−0.323 − 0.235i)23-s + (−0.444 + 0.323i)25-s + 5.19i·27-s + (1.59 − 0.516i)29-s + (0.969 − 2.98i)31-s + ⋯
L(s)  = 1  + 0.617i·3-s + (0.325 + 1.00i)5-s + (−0.222 − 0.305i)7-s + 0.618·9-s + (−1.52 − 0.495i)11-s + (−1.04 + 1.44i)13-s + (−0.618 + 0.200i)15-s + (−1.66 − 0.542i)17-s + (−0.677 − 0.933i)19-s + (0.188 − 0.137i)21-s + (−0.0674 − 0.0490i)23-s + (−0.0889 + 0.0646i)25-s + 0.999i·27-s + (0.295 − 0.0959i)29-s + (0.174 − 0.536i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 + 0.279i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.960 + 0.279i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.960 + 0.279i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ -0.960 + 0.279i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4856691187\)
\(L(\frac12)\) \(\approx\) \(0.4856691187\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (0.587 + 0.809i)T \)
41 \( 1 + (-1.37 - 6.25i)T \)
good3 \( 1 - 1.06iT - 3T^{2} \)
5 \( 1 + (-0.727 - 2.24i)T + (-4.04 + 2.93i)T^{2} \)
11 \( 1 + (5.06 + 1.64i)T + (8.89 + 6.46i)T^{2} \)
13 \( 1 + (3.77 - 5.19i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (6.88 + 2.23i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (2.95 + 4.06i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + (0.323 + 0.235i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-1.59 + 0.516i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (-0.969 + 2.98i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-1.22 - 3.75i)T + (-29.9 + 21.7i)T^{2} \)
43 \( 1 + (-2.21 - 1.61i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (4.48 - 6.16i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-7.71 + 2.50i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (8.99 + 6.53i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (10.7 - 7.80i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-12.1 + 3.93i)T + (54.2 - 39.3i)T^{2} \)
71 \( 1 + (10.8 + 3.53i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + 13.1T + 73T^{2} \)
79 \( 1 - 11.9iT - 79T^{2} \)
83 \( 1 - 14.5T + 83T^{2} \)
89 \( 1 + (8.48 + 11.6i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (5.79 - 1.88i)T + (78.4 - 57.0i)T^{2} \)
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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

−10.21926088435951323577424155179, −9.610680827690645043451893979188, −8.826392086056390074897170064010, −7.58580175159840578181981210006, −6.87755060811371319427777962642, −6.27882278831174457862908003066, −4.73636948191129252035376827715, −4.46540324525997889306025342099, −2.91522597835281623631843657296, −2.27044117891342096601633308540, 0.19022103459018541924309614302, 1.82552216562474733401857696593, 2.65608611394415823859804290666, 4.30878338357019753466708850322, 5.11697854779801014682603133435, 5.85185027224100302633065063325, 6.97465328098842240168478730659, 7.77409113129678375307330990445, 8.415232609972085688327317732049, 9.298874201669924621186308834980

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