Properties

Label 2-1148-287.163-c1-0-3
Degree $2$
Conductor $1148$
Sign $-0.631 - 0.775i$
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  + (−0.388 + 0.224i)3-s + (−0.254 + 0.440i)5-s + (2.49 + 0.888i)7-s + (−1.39 + 2.42i)9-s + (−0.882 + 0.509i)11-s + 1.40i·13-s − 0.228i·15-s + (−1.94 + 1.12i)17-s + (−4.34 − 2.50i)19-s + (−1.16 + 0.213i)21-s + (−1.64 + 2.84i)23-s + (2.37 + 4.10i)25-s − 2.60i·27-s − 6.91i·29-s + (1.54 + 2.67i)31-s + ⋯
L(s)  = 1  + (−0.224 + 0.129i)3-s + (−0.113 + 0.197i)5-s + (0.941 + 0.335i)7-s + (−0.466 + 0.807i)9-s + (−0.266 + 0.153i)11-s + 0.390i·13-s − 0.0589i·15-s + (−0.470 + 0.271i)17-s + (−0.995 − 0.574i)19-s + (−0.254 + 0.0466i)21-s + (−0.343 + 0.594i)23-s + (0.474 + 0.821i)25-s − 0.500i·27-s − 1.28i·29-s + (0.277 + 0.481i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.631 - 0.775i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.631 - 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.631 - 0.775i$
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: \((2,\ 1148,\ (\ :1/2),\ -0.631 - 0.775i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9941744183\)
\(L(\frac12)\) \(\approx\) \(0.9941744183\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.49 - 0.888i)T \)
41 \( 1 + (2.71 + 5.79i)T \)
good3 \( 1 + (0.388 - 0.224i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.254 - 0.440i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.882 - 0.509i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.40iT - 13T^{2} \)
17 \( 1 + (1.94 - 1.12i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.34 + 2.50i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.64 - 2.84i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.91iT - 29T^{2} \)
31 \( 1 + (-1.54 - 2.67i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.57 - 6.19i)T + (-18.5 - 32.0i)T^{2} \)
43 \( 1 + 4.41T + 43T^{2} \)
47 \( 1 + (-2.96 - 1.71i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.47 - 3.16i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.98 - 5.17i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.80 - 10.0i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.50 - 3.75i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.32iT - 71T^{2} \)
73 \( 1 + (2.33 + 4.04i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-10.0 - 5.77i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 9.00T + 83T^{2} \)
89 \( 1 + (5.78 + 3.33i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 5.65iT - 97T^{2} \)
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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

−10.35018414000282655320009797543, −9.130782439617640328549588684809, −8.436371623821077309286675428448, −7.74053992174106309168441888400, −6.78585317133967028709763227820, −5.75683254579094710272993564482, −4.94241645664392306076611981920, −4.20363360099851474012730724637, −2.71689068101020961869914589462, −1.76979241451396188067179593579, 0.42906004398912897466949555551, 1.89861109943322539127343881662, 3.25906993198403975104183948111, 4.36445142473675945506464819790, 5.16964639590268328883026889417, 6.19084797053139968673786657864, 6.92701407963724383343031640730, 8.092779657815832955202655160738, 8.489547873457108915215040918329, 9.435645596249086229596230272484

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