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Properties

Label 2-1148-41.10-c1-0-5
Degree $2$
Conductor $1148$
Sign $-0.00530 - 0.999i$
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  + 0.327·3-s + (0.570 + 1.75i)5-s + (0.809 − 0.587i)7-s − 2.89·9-s + (−0.300 + 0.923i)11-s + (2.31 + 1.68i)13-s + (0.186 + 0.575i)15-s + (−0.414 + 1.27i)17-s + (−3.94 + 2.86i)19-s + (0.265 − 0.192i)21-s + (5.85 + 4.25i)23-s + (1.28 − 0.936i)25-s − 1.93·27-s + (−0.780 − 2.40i)29-s + (−2.69 + 8.29i)31-s + ⋯
L(s)  = 1  + 0.189·3-s + (0.255 + 0.784i)5-s + (0.305 − 0.222i)7-s − 0.964·9-s + (−0.0905 + 0.278i)11-s + (0.642 + 0.466i)13-s + (0.0482 + 0.148i)15-s + (−0.100 + 0.309i)17-s + (−0.904 + 0.657i)19-s + (0.0578 − 0.0420i)21-s + (1.22 + 0.887i)23-s + (0.257 − 0.187i)25-s − 0.371·27-s + (−0.144 − 0.445i)29-s + (−0.484 + 1.49i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00530 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00530 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.00530 - 0.999i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (953, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ -0.00530 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.510752746\)
\(L(\frac12)\) \(\approx\) \(1.510752746\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.809 + 0.587i)T \)
41 \( 1 + (6.03 + 2.13i)T \)
good3 \( 1 - 0.327T + 3T^{2} \)
5 \( 1 + (-0.570 - 1.75i)T + (-4.04 + 2.93i)T^{2} \)
11 \( 1 + (0.300 - 0.923i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-2.31 - 1.68i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.414 - 1.27i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (3.94 - 2.86i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-5.85 - 4.25i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (0.780 + 2.40i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (2.69 - 8.29i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-1.80 - 5.54i)T + (-29.9 + 21.7i)T^{2} \)
43 \( 1 + (2.05 + 1.49i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (-8.41 - 6.11i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-1.78 - 5.48i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (9.80 + 7.12i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-4.15 + 3.01i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-2.83 - 8.71i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-0.614 + 1.89i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + 8.52T + 73T^{2} \)
79 \( 1 + 14.0T + 79T^{2} \)
83 \( 1 - 8.76T + 83T^{2} \)
89 \( 1 + (-7.09 + 5.15i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-3.86 - 11.8i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.16208820613600465136585295192, −9.019290537865699439640106563158, −8.505292247951959768615611818992, −7.49999471737938302672564937196, −6.64639066653014989020360579368, −5.92063228868846063805539843679, −4.87597313066373169699345684622, −3.69145541852816118283633371345, −2.81771329593485932055974802145, −1.62067960600630623229448393567, 0.64288436857273876423077857034, 2.19872422796641666883258624987, 3.22615899312953657173315175558, 4.50000727718862159627659841605, 5.35923677869087072705992149976, 6.04281659881009990883839836257, 7.15033128415027131956227988364, 8.260583397592155174860423288202, 8.800719994936992427705676206173, 9.235669502474130241902616879077

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