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Properties

Label 2-1148-1148.163-c0-0-4
Degree $2$
Conductor $1148$
Sign $-0.660 + 0.750i$
Analytic cond. $0.572926$
Root an. cond. $0.756919$
Motivic weight $0$
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)2-s + (−0.258 − 0.448i)3-s + (−0.499 − 0.866i)4-s + (−0.866 + 1.5i)5-s − 0.517·6-s + (0.965 − 0.258i)7-s − 0.999·8-s + (0.366 − 0.633i)9-s + (0.866 + 1.5i)10-s + (−0.965 − 1.67i)11-s + (−0.258 + 0.448i)12-s + (0.258 − 0.965i)14-s + 0.896·15-s + (−0.5 + 0.866i)16-s + (−0.366 − 0.633i)18-s + (0.965 − 1.67i)19-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.258 − 0.448i)3-s + (−0.499 − 0.866i)4-s + (−0.866 + 1.5i)5-s − 0.517·6-s + (0.965 − 0.258i)7-s − 0.999·8-s + (0.366 − 0.633i)9-s + (0.866 + 1.5i)10-s + (−0.965 − 1.67i)11-s + (−0.258 + 0.448i)12-s + (0.258 − 0.965i)14-s + 0.896·15-s + (−0.5 + 0.866i)16-s + (−0.366 − 0.633i)18-s + (0.965 − 1.67i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.660 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.660 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.660 + 0.750i$
Analytic conductor: \(0.572926\)
Root analytic conductor: \(0.756919\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :0),\ -0.660 + 0.750i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9980450020\)
\(L(\frac12)\) \(\approx\) \(0.9980450020\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-0.965 + 0.258i)T \)
41 \( 1 - T \)
good3 \( 1 + (0.258 + 0.448i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.965 + 1.67i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.965 + 1.67i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - 0.517T + T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.258 - 0.448i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−10.09125481826192331939549194678, −8.943472275105697750196999862961, −7.920697713057303702607434151905, −7.18986565058325321158550631745, −6.29979458188865015541155664449, −5.38564920077169373858192926865, −4.24946721618541658633344012891, −3.26727301950420158085170848470, −2.63367365765967059065735941983, −0.832651951057402292297510188007, 1.87003989497706340214766953971, 3.80321237972504559655340213589, 4.65467278192948752454025170247, 4.98543577967690046327758738586, 5.64197311664485540424236186105, 7.35092087927488955548503945285, 7.83335761901023533582999590133, 8.264756880022977805897634672719, 9.362226675572293137132348782339, 10.09617549833778174512855795706

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