Properties

Label 2-1148-41.16-c1-0-7
Degree $2$
Conductor $1148$
Sign $0.985 + 0.170i$
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  − 2.87·3-s + (2.43 + 1.77i)5-s + (−0.309 − 0.951i)7-s + 5.25·9-s + (−0.220 + 0.160i)11-s + (0.746 − 2.29i)13-s + (−7.01 − 5.09i)15-s + (2.08 − 1.51i)17-s + (−0.484 − 1.49i)19-s + (0.887 + 2.73i)21-s + (−0.540 + 1.66i)23-s + (1.26 + 3.89i)25-s − 6.48·27-s + (−4.86 − 3.53i)29-s + (−1.36 + 0.991i)31-s + ⋯
L(s)  = 1  − 1.65·3-s + (1.09 + 0.792i)5-s + (−0.116 − 0.359i)7-s + 1.75·9-s + (−0.0666 + 0.0484i)11-s + (0.206 − 0.636i)13-s + (−1.81 − 1.31i)15-s + (0.504 − 0.366i)17-s + (−0.111 − 0.342i)19-s + (0.193 + 0.596i)21-s + (−0.112 + 0.346i)23-s + (0.253 + 0.778i)25-s − 1.24·27-s + (−0.903 − 0.656i)29-s + (−0.245 + 0.178i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.170i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 + 0.170i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.985 + 0.170i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (57, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ 0.985 + 0.170i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.083718908\)
\(L(\frac12)\) \(\approx\) \(1.083718908\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.309 + 0.951i)T \)
41 \( 1 + (-2.41 + 5.93i)T \)
good3 \( 1 + 2.87T + 3T^{2} \)
5 \( 1 + (-2.43 - 1.77i)T + (1.54 + 4.75i)T^{2} \)
11 \( 1 + (0.220 - 0.160i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-0.746 + 2.29i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-2.08 + 1.51i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.484 + 1.49i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (0.540 - 1.66i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (4.86 + 3.53i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.36 - 0.991i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-6.90 - 5.01i)T + (11.4 + 35.1i)T^{2} \)
43 \( 1 + (0.500 - 1.53i)T + (-34.7 - 25.2i)T^{2} \)
47 \( 1 + (-1.47 + 4.55i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-7.63 - 5.54i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.15 + 6.64i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (1.03 + 3.17i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-6.99 - 5.08i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (-10.3 + 7.54i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 - 12.6T + 73T^{2} \)
79 \( 1 - 9.64T + 79T^{2} \)
83 \( 1 - 5.26T + 83T^{2} \)
89 \( 1 + (3.72 + 11.4i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (4.44 + 3.23i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.00864981471534244419787736796, −9.397743473071280107087954412025, −7.85284959481987232205121103180, −6.97502936372355500570883506403, −6.28255505334430880438171637632, −5.66363036495336833836883310565, −4.97391345298219008111690213176, −3.66416410235445477585512809324, −2.26267506105874569884501595547, −0.78005946177275463413468879292, 0.986347691030792174511442709700, 2.08255277889895015029336282039, 3.98522112367976219380375331605, 5.00904813617901851558569744409, 5.66529104023745405245000809462, 6.12886177028655889409226247159, 7.01401901489652669170335139191, 8.243953362908193904123579340316, 9.332882055709664809315276057003, 9.778556579285749606461955395941

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