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Properties

Label 2-1148-41.32-c1-0-18
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: $\chi_{1148} (729, \cdot )$
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

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

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