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Properties

Label 2-1148-7.2-c1-0-1
Degree $2$
Conductor $1148$
Sign $0.636 - 0.770i$
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  + (−1.45 − 2.51i)3-s + (−0.253 + 0.438i)5-s + (−2.53 + 0.764i)7-s + (−2.72 + 4.71i)9-s + (−2.85 − 4.94i)11-s + 2.51·13-s + 1.47·15-s + (−2.51 − 4.35i)17-s + (−2.89 + 5.00i)19-s + (5.60 + 5.26i)21-s + (−0.759 + 1.31i)23-s + (2.37 + 4.10i)25-s + 7.11·27-s + 6.63·29-s + (1.63 + 2.83i)31-s + ⋯
L(s)  = 1  + (−0.838 − 1.45i)3-s + (−0.113 + 0.196i)5-s + (−0.957 + 0.289i)7-s + (−0.907 + 1.57i)9-s + (−0.860 − 1.49i)11-s + 0.697·13-s + 0.379·15-s + (−0.610 − 1.05i)17-s + (−0.663 + 1.14i)19-s + (1.22 + 1.14i)21-s + (−0.158 + 0.274i)23-s + (0.474 + 0.821i)25-s + 1.36·27-s + 1.23·29-s + (0.294 + 0.509i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.636 - 0.770i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.636 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.636 - 0.770i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (821, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ 0.636 - 0.770i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3431153959\)
\(L(\frac12)\) \(\approx\) \(0.3431153959\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.53 - 0.764i)T \)
41 \( 1 + T \)
good3 \( 1 + (1.45 + 2.51i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.253 - 0.438i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.85 + 4.94i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.51T + 13T^{2} \)
17 \( 1 + (2.51 + 4.35i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.89 - 5.00i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.759 - 1.31i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.63T + 29T^{2} \)
31 \( 1 + (-1.63 - 2.83i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.91 - 8.51i)T + (-18.5 - 32.0i)T^{2} \)
43 \( 1 + 8.37T + 43T^{2} \)
47 \( 1 + (-6.10 + 10.5i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.05 - 10.4i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.39 - 7.61i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.03 + 5.26i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.60 - 9.71i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.46T + 71T^{2} \)
73 \( 1 + (2.42 + 4.20i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.09 + 1.90i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 0.906T + 83T^{2} \)
89 \( 1 + (-2.51 + 4.35i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.21T + 97T^{2} \)
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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

−10.22669549644246043181382014763, −8.722960595778640297598388124311, −8.306406195382997512827838494737, −7.17137694317494013663816389196, −6.56808114243132614314360958557, −5.90103591498610498306959296625, −5.20649430504608473169839601010, −3.45582778360634306376080854751, −2.55523081647436183361906406719, −1.05490639553968038443844218149, 0.19477498336169026643845584534, 2.49874593565588784629864786859, 3.87121175386865658707754580908, 4.44239401458164662168799544499, 5.20464185352420366368488847318, 6.30892069119424729301162337503, 6.86546461840858980149292524689, 8.286431597264056600589976412470, 9.103155372600795643879869088019, 9.921429709082668830519678361755

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