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Properties

Label 2-1148-7.4-c1-0-15
Degree $2$
Conductor $1148$
Sign $0.940 + 0.340i$
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  + (−1.36 + 2.37i)3-s + (0.431 + 0.747i)5-s + (−2.57 + 0.627i)7-s + (−2.24 − 3.89i)9-s + (2.79 − 4.84i)11-s − 1.90·13-s − 2.36·15-s + (2.00 − 3.48i)17-s + (0.958 + 1.66i)19-s + (2.02 − 6.95i)21-s + (−4.53 − 7.85i)23-s + (2.12 − 3.68i)25-s + 4.08·27-s + 5.61·29-s + (−4.55 + 7.89i)31-s + ⋯
L(s)  = 1  + (−0.790 + 1.36i)3-s + (0.192 + 0.334i)5-s + (−0.971 + 0.237i)7-s + (−0.748 − 1.29i)9-s + (0.842 − 1.45i)11-s − 0.526·13-s − 0.610·15-s + (0.487 − 0.844i)17-s + (0.219 + 0.380i)19-s + (0.442 − 1.51i)21-s + (−0.945 − 1.63i)23-s + (0.425 − 0.737i)25-s + 0.786·27-s + 1.04·29-s + (−0.818 + 1.41i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 + 0.340i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.940 + 0.340i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.940 + 0.340i$
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: \((2,\ 1148,\ (\ :1/2),\ 0.940 + 0.340i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8037154999\)
\(L(\frac12)\) \(\approx\) \(0.8037154999\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.57 - 0.627i)T \)
41 \( 1 + T \)
good3 \( 1 + (1.36 - 2.37i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.431 - 0.747i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.79 + 4.84i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.90T + 13T^{2} \)
17 \( 1 + (-2.00 + 3.48i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.958 - 1.66i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.53 + 7.85i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 5.61T + 29T^{2} \)
31 \( 1 + (4.55 - 7.89i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.728 + 1.26i)T + (-18.5 + 32.0i)T^{2} \)
43 \( 1 - 7.83T + 43T^{2} \)
47 \( 1 + (5.08 + 8.79i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.582 + 1.00i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.43 - 5.94i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.61 + 4.52i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.32 - 12.6i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.14T + 71T^{2} \)
73 \( 1 + (-2.20 + 3.82i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.65 - 4.59i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6.73T + 83T^{2} \)
89 \( 1 + (-1.99 - 3.46i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 3.44T + 97T^{2} \)
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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

−10.01552629702775406260479277443, −9.100074797984950782973784566907, −8.484407119485685895163999347514, −6.92011637001760826373015141130, −6.20048895147356070116267664639, −5.57094114469711012764800799884, −4.56483649060600704892051546095, −3.59277087535558963810906607398, −2.82876466621432957847038279914, −0.43604715926975042277745835855, 1.18555404313905319619091266630, 2.11160969911455948726451844781, 3.66066657755289820932806903072, 4.85970178038650333632432808294, 5.94613253859060535313269579824, 6.44737175774941485243695649352, 7.41879219910186664422481520252, 7.68827413413230029059162788140, 9.305252184391461513909423537722, 9.656727956413049562444496050367

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