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Properties

Label 2-1148-41.16-c1-0-13
Degree $2$
Conductor $1148$
Sign $0.996 + 0.0835i$
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  + 2.26·3-s + (1.20 + 0.877i)5-s + (−0.309 − 0.951i)7-s + 2.14·9-s + (3.91 − 2.84i)11-s + (−0.142 + 0.437i)13-s + (2.73 + 1.99i)15-s + (1.61 − 1.17i)17-s + (0.659 + 2.03i)19-s + (−0.700 − 2.15i)21-s + (−1.05 + 3.23i)23-s + (−0.856 − 2.63i)25-s − 1.93·27-s + (1.34 + 0.975i)29-s + (3.63 − 2.64i)31-s + ⋯
L(s)  = 1  + 1.30·3-s + (0.540 + 0.392i)5-s + (−0.116 − 0.359i)7-s + 0.715·9-s + (1.18 − 0.858i)11-s + (−0.0394 + 0.121i)13-s + (0.707 + 0.513i)15-s + (0.392 − 0.285i)17-s + (0.151 + 0.465i)19-s + (−0.152 − 0.470i)21-s + (−0.219 + 0.674i)23-s + (−0.171 − 0.527i)25-s − 0.373·27-s + (0.249 + 0.181i)29-s + (0.652 − 0.474i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0835i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.996 + 0.0835i$
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: \((2,\ 1148,\ (\ :1/2),\ 0.996 + 0.0835i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.903728414\)
\(L(\frac12)\) \(\approx\) \(2.903728414\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.88 + 6.11i)T \)
good3 \( 1 - 2.26T + 3T^{2} \)
5 \( 1 + (-1.20 - 0.877i)T + (1.54 + 4.75i)T^{2} \)
11 \( 1 + (-3.91 + 2.84i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (0.142 - 0.437i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.61 + 1.17i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.659 - 2.03i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (1.05 - 3.23i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-1.34 - 0.975i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-3.63 + 2.64i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.968 + 0.704i)T + (11.4 + 35.1i)T^{2} \)
43 \( 1 + (2.34 - 7.21i)T + (-34.7 - 25.2i)T^{2} \)
47 \( 1 + (2.75 - 8.48i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-7.81 - 5.68i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-1.89 + 5.82i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-1.05 - 3.24i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (6.57 + 4.77i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (0.586 - 0.426i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + 3.52T + 73T^{2} \)
79 \( 1 + 7.17T + 79T^{2} \)
83 \( 1 + 6.91T + 83T^{2} \)
89 \( 1 + (-1.92 - 5.93i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-2.29 - 1.67i)T + (29.9 + 92.2i)T^{2} \)
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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

−9.613313603186242333387309289918, −9.044830034091475190854782334872, −8.212404358858038480715098376847, −7.48056334657639200453548987338, −6.47630406525159992537607247862, −5.74749211358324705682216612736, −4.23137622990884154104416574298, −3.44217686060920577114949416682, −2.61439905574779179674289260875, −1.36412482475493721819227448961, 1.50885012324380529524416570856, 2.44803816024822752580165276375, 3.50102036351634007778742361551, 4.45520936697948856120537112864, 5.54319953372693942513445211427, 6.61274759631118431394265023199, 7.41059663302549609161974884092, 8.538705593643231496817359479442, 8.830206521417868294687363240164, 9.729540927897645328688973478232

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