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Properties

Label 2-1148-287.163-c1-0-4
Degree $2$
Conductor $1148$
Sign $-0.957 - 0.289i$
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  + (−2.68 + 1.54i)3-s + (0.0658 − 0.114i)5-s + (−0.113 + 2.64i)7-s + (3.28 − 5.69i)9-s + (0.855 − 0.493i)11-s + 3.93i·13-s + 0.407i·15-s + (0.359 − 0.207i)17-s + (6.06 + 3.50i)19-s + (−3.78 − 7.25i)21-s + (−2.16 + 3.74i)23-s + (2.49 + 4.31i)25-s + 11.0i·27-s − 8.82i·29-s + (2.99 + 5.19i)31-s + ⋯
L(s)  = 1  + (−1.54 + 0.893i)3-s + (0.0294 − 0.0510i)5-s + (−0.0427 + 0.999i)7-s + (1.09 − 1.89i)9-s + (0.257 − 0.148i)11-s + 1.09i·13-s + 0.105i·15-s + (0.0871 − 0.0503i)17-s + (1.39 + 0.803i)19-s + (−0.826 − 1.58i)21-s + (−0.451 + 0.781i)23-s + (0.498 + 0.863i)25-s + 2.13i·27-s − 1.63i·29-s + (0.538 + 0.932i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.289i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.957 - 0.289i$
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: \((2,\ 1148,\ (\ :1/2),\ -0.957 - 0.289i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7063591298\)
\(L(\frac12)\) \(\approx\) \(0.7063591298\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.113 - 2.64i)T \)
41 \( 1 + (4.75 - 4.28i)T \)
good3 \( 1 + (2.68 - 1.54i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.0658 + 0.114i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.855 + 0.493i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.93iT - 13T^{2} \)
17 \( 1 + (-0.359 + 0.207i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-6.06 - 3.50i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.16 - 3.74i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 8.82iT - 29T^{2} \)
31 \( 1 + (-2.99 - 5.19i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.35 + 4.07i)T + (-18.5 - 32.0i)T^{2} \)
43 \( 1 - 4.65T + 43T^{2} \)
47 \( 1 + (8.79 + 5.07i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (11.6 - 6.74i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.79 - 6.57i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.86 + 6.70i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (8.36 - 4.82i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.67iT - 71T^{2} \)
73 \( 1 + (4.42 + 7.66i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (13.4 + 7.75i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 4.23T + 83T^{2} \)
89 \( 1 + (1.66 + 0.959i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 12.6iT - 97T^{2} \)
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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

−10.02113563936552638802552171030, −9.617853755906189524862961707040, −8.859894812393166158305696010966, −7.57500536914850268042640724622, −6.45971718733307034229700418871, −5.85356950497839610787624758342, −5.17335194902297184674958612386, −4.32180461894670179630059097732, −3.26965012920676838411679677896, −1.46199682538939290210820039855, 0.43355328555648565679718768011, 1.35823018738225425984966897212, 3.03998866167460081853925752686, 4.53642432691646911634491130346, 5.21314109768823515708767060507, 6.18604968165528579609443059204, 6.86679591859841412283132822451, 7.49468497176850279887940868875, 8.311068230337091783065224974935, 9.814362327110416223070243494069

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