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Properties

Label 2-1148-7.4-c1-0-14
Degree $2$
Conductor $1148$
Sign $0.324 - 0.945i$
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  + (−0.370 + 0.642i)3-s + (1.24 + 2.15i)5-s + (0.670 + 2.55i)7-s + (1.22 + 2.12i)9-s + (2.66 − 4.61i)11-s + 4.45·13-s − 1.84·15-s + (2.61 − 4.52i)17-s + (−0.986 − 1.70i)19-s + (−1.89 − 0.518i)21-s + (3.48 + 6.04i)23-s + (−0.591 + 1.02i)25-s − 4.04·27-s + 3.94·29-s + (−2.01 + 3.49i)31-s + ⋯
L(s)  = 1  + (−0.214 + 0.370i)3-s + (0.556 + 0.963i)5-s + (0.253 + 0.967i)7-s + (0.408 + 0.707i)9-s + (0.803 − 1.39i)11-s + 1.23·13-s − 0.475·15-s + (0.633 − 1.09i)17-s + (−0.226 − 0.392i)19-s + (−0.412 − 0.113i)21-s + (0.727 + 1.25i)23-s + (−0.118 + 0.204i)25-s − 0.777·27-s + 0.733·29-s + (−0.362 + 0.628i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.324 - 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.324 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.324 - 0.945i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ 0.324 - 0.945i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.010674503\)
\(L(\frac12)\) \(\approx\) \(2.010674503\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.670 - 2.55i)T \)
41 \( 1 + T \)
good3 \( 1 + (0.370 - 0.642i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.24 - 2.15i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.66 + 4.61i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 4.45T + 13T^{2} \)
17 \( 1 + (-2.61 + 4.52i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.986 + 1.70i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.48 - 6.04i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.94T + 29T^{2} \)
31 \( 1 + (2.01 - 3.49i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.22 + 7.30i)T + (-18.5 + 32.0i)T^{2} \)
43 \( 1 + 10.5T + 43T^{2} \)
47 \( 1 + (-2.33 - 4.04i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.07 - 1.86i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.53 - 2.65i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.01 - 10.4i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.81 + 13.5i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.549T + 71T^{2} \)
73 \( 1 + (-3.05 + 5.28i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.28 + 12.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 16.2T + 83T^{2} \)
89 \( 1 + (1.51 + 2.61i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 11.3T + 97T^{2} \)
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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

−10.02416877454232256191372689628, −9.098097380237539191530949238285, −8.555384470260714228315457677412, −7.39998448947284564265074475994, −6.46977595592940875349837984155, −5.73534160734945298411973303588, −5.04633830000563024372652857093, −3.59100709130272045483980457347, −2.84288108496579770290509596045, −1.48002887520008870339216811026, 1.14001055403663323111040418619, 1.63453234207946964961414097746, 3.70863937769688041342573659203, 4.34398046546920460181826774337, 5.35239981825342419031847904187, 6.58795471382800259331097673015, 6.80531491904409395581910639788, 8.158096674472717962320649539691, 8.720603890045518959698444267186, 9.827826418265931061923171462559

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