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Properties

Label 2-1148-287.163-c1-0-1
Degree $2$
Conductor $1148$
Sign $-0.986 - 0.162i$
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  + (−1.64 + 0.946i)3-s + (1.83 − 3.17i)5-s + (0.288 + 2.63i)7-s + (0.293 − 0.507i)9-s + (−4.51 + 2.60i)11-s − 3.89i·13-s + 6.94i·15-s + (1.07 − 0.618i)17-s + (−3.64 − 2.10i)19-s + (−2.96 − 4.04i)21-s + (−1.39 + 2.41i)23-s + (−4.21 − 7.30i)25-s − 4.57i·27-s + 4.81i·29-s + (4.20 + 7.29i)31-s + ⋯
L(s)  = 1  + (−0.946 + 0.546i)3-s + (0.819 − 1.41i)5-s + (0.108 + 0.994i)7-s + (0.0977 − 0.169i)9-s + (−1.36 + 0.786i)11-s − 1.08i·13-s + 1.79i·15-s + (0.259 − 0.149i)17-s + (−0.836 − 0.483i)19-s + (−0.646 − 0.881i)21-s + (−0.290 + 0.503i)23-s + (−0.842 − 1.46i)25-s − 0.879i·27-s + 0.893i·29-s + (0.755 + 1.30i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.162i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.986 - 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.986 - 0.162i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ -0.986 - 0.162i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1897405414\)
\(L(\frac12)\) \(\approx\) \(0.1897405414\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.288 - 2.63i)T \)
41 \( 1 + (4.85 - 4.17i)T \)
good3 \( 1 + (1.64 - 0.946i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.83 + 3.17i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (4.51 - 2.60i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 3.89iT - 13T^{2} \)
17 \( 1 + (-1.07 + 0.618i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.64 + 2.10i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.39 - 2.41i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.81iT - 29T^{2} \)
31 \( 1 + (-4.20 - 7.29i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.952 - 1.64i)T + (-18.5 - 32.0i)T^{2} \)
43 \( 1 + 11.5T + 43T^{2} \)
47 \( 1 + (1.91 + 1.10i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.05 - 1.18i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.14 + 10.6i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.31 - 2.27i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.49 + 2.01i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 11.2iT - 71T^{2} \)
73 \( 1 + (-7.92 - 13.7i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.10 + 2.94i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 17.1T + 83T^{2} \)
89 \( 1 + (1.77 + 1.02i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 2.27iT - 97T^{2} \)
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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

−10.12273417336198960572268867898, −9.585179698183194785267566516119, −8.472027273501608747360256426130, −8.106108305948340143753114433070, −6.54824491723910997541279258565, −5.54517539793500617754003113389, −5.07504541006420822079021625782, −4.80829804218200376407698077876, −2.88558310122793684210931379407, −1.68198329174628656795465309729, 0.087107727830945077481983308674, 1.82615411958634052521507960076, 2.93995566396902786381565664132, 4.13987180487308682880765054694, 5.46328714420231911725936846669, 6.24857687274411240053240801646, 6.66591885602982706494461703736, 7.51264009383455456001486923987, 8.402645835240685772931652950895, 9.886607359569101236756301309140

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