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Properties

Label 2-1148-7.4-c1-0-7
Degree $2$
Conductor $1148$
Sign $-0.605 - 0.795i$
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  + (−0.5 + 0.866i)3-s + (1.5 + 2.59i)5-s + (2.5 + 0.866i)7-s + (1 + 1.73i)9-s + (−1.5 + 2.59i)11-s − 4·13-s − 3·15-s + (3.5 + 6.06i)19-s + (−2 + 1.73i)21-s + (−3 − 5.19i)23-s + (−2 + 3.46i)25-s − 5·27-s + 6·29-s + (5 − 8.66i)31-s + (−1.5 − 2.59i)33-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (0.670 + 1.16i)5-s + (0.944 + 0.327i)7-s + (0.333 + 0.577i)9-s + (−0.452 + 0.783i)11-s − 1.10·13-s − 0.774·15-s + (0.802 + 1.39i)19-s + (−0.436 + 0.377i)21-s + (−0.625 − 1.08i)23-s + (−0.400 + 0.692i)25-s − 0.962·27-s + 1.11·29-s + (0.898 − 1.55i)31-s + (−0.261 − 0.452i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.605 - 0.795i$
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: \((2,\ 1148,\ (\ :1/2),\ -0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.673943899\)
\(L(\frac12)\) \(\approx\) \(1.673943899\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 - 0.866i)T \)
41 \( 1 + T \)
good3 \( 1 + (0.5 - 0.866i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.5 - 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (-5 + 8.66i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (6 + 10.3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3 - 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.5 - 11.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 9T + 71T^{2} \)
73 \( 1 + (7 - 12.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + (-6 - 10.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−10.19106632542635086340634989061, −9.745729731948265619225177278201, −8.270633291709258339055474741293, −7.63486063245304789731337065657, −6.78910354742494506476077676976, −5.73073252471695500013273618058, −5.00204516769489710156568694167, −4.16636081779854867182988028584, −2.59876736114822818325043109856, −1.98504048598254542901029208621, 0.77575176012664874582822881449, 1.65635820597520950498015914372, 3.10841682492313998810225076496, 4.76953954725935704611228466520, 5.00807182546260997785011896876, 6.07925906365315351098141303553, 7.06770172590634087964419016770, 7.87167810182071626439439879439, 8.698030150360916937617584052215, 9.480014696826783134729229895965

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