Properties

Label 2-1148-41.9-c1-0-0
Degree $2$
Conductor $1148$
Sign $-0.967 - 0.254i$
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  + (−0.898 − 0.898i)3-s + 3.17i·5-s + (−0.707 − 0.707i)7-s − 1.38i·9-s + (−3.23 − 3.23i)11-s + (3.60 + 3.60i)13-s + (2.85 − 2.85i)15-s + (−4.90 + 4.90i)17-s + (1.76 − 1.76i)19-s + 1.27i·21-s + 7.82·23-s − 5.08·25-s + (−3.94 + 3.94i)27-s + (−1.94 − 1.94i)29-s − 10.1·31-s + ⋯
L(s)  = 1  + (−0.518 − 0.518i)3-s + 1.42i·5-s + (−0.267 − 0.267i)7-s − 0.461i·9-s + (−0.974 − 0.974i)11-s + (0.999 + 0.999i)13-s + (0.736 − 0.736i)15-s + (−1.18 + 1.18i)17-s + (0.405 − 0.405i)19-s + 0.277i·21-s + 1.63·23-s − 1.01·25-s + (−0.758 + 0.758i)27-s + (−0.360 − 0.360i)29-s − 1.82·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 - 0.254i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.967 - 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.967 - 0.254i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ -0.967 - 0.254i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1779500755\)
\(L(\frac12)\) \(\approx\) \(0.1779500755\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 + 0.707i)T \)
41 \( 1 + (6.28 - 1.21i)T \)
good3 \( 1 + (0.898 + 0.898i)T + 3iT^{2} \)
5 \( 1 - 3.17iT - 5T^{2} \)
11 \( 1 + (3.23 + 3.23i)T + 11iT^{2} \)
13 \( 1 + (-3.60 - 3.60i)T + 13iT^{2} \)
17 \( 1 + (4.90 - 4.90i)T - 17iT^{2} \)
19 \( 1 + (-1.76 + 1.76i)T - 19iT^{2} \)
23 \( 1 - 7.82T + 23T^{2} \)
29 \( 1 + (1.94 + 1.94i)T + 29iT^{2} \)
31 \( 1 + 10.1T + 31T^{2} \)
37 \( 1 + 10.4T + 37T^{2} \)
43 \( 1 + 2.44iT - 43T^{2} \)
47 \( 1 + (3.19 - 3.19i)T - 47iT^{2} \)
53 \( 1 + (0.796 + 0.796i)T + 53iT^{2} \)
59 \( 1 + 11.7T + 59T^{2} \)
61 \( 1 + 14.3iT - 61T^{2} \)
67 \( 1 + (4.45 - 4.45i)T - 67iT^{2} \)
71 \( 1 + (-0.431 - 0.431i)T + 71iT^{2} \)
73 \( 1 - 6.43iT - 73T^{2} \)
79 \( 1 + (-0.490 - 0.490i)T + 79iT^{2} \)
83 \( 1 + 2.68T + 83T^{2} \)
89 \( 1 + (-9.23 - 9.23i)T + 89iT^{2} \)
97 \( 1 + (-5.19 + 5.19i)T - 97iT^{2} \)
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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

−10.57529773459681355731809646607, −9.284098762988989928568482514194, −8.593904863307954951113017271826, −7.38285671210701331938443606588, −6.69158307600307536052360311846, −6.32474519962054706026797542859, −5.31673522833926155395576505101, −3.74831176898676959955963063550, −3.15018081902989461931239208865, −1.73055169847156199870491807570, 0.079772762781410226716436458609, 1.72744050921880231683524573546, 3.18089360029461757766326763143, 4.54248869522268315921038078435, 5.20566166591039923199012569147, 5.50798794308820629431637275893, 7.01088194955991139127763185496, 7.84127824323511355347492004660, 8.862204570379789258376598830216, 9.246472956744590775522764788512

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