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Properties

Label 2-1148-287.163-c1-0-25
Degree $2$
Conductor $1148$
Sign $-0.0150 + 0.999i$
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  + (1.64 − 0.946i)3-s + (1.83 − 3.17i)5-s + (−0.288 − 2.63i)7-s + (0.293 − 0.507i)9-s + (4.51 − 2.60i)11-s + 3.89i·13-s − 6.94i·15-s + (−1.07 + 0.618i)17-s + (3.64 + 2.10i)19-s + (−2.96 − 4.04i)21-s + (−1.39 + 2.41i)23-s + (−4.21 − 7.30i)25-s + 4.57i·27-s − 4.81i·29-s + (4.20 + 7.29i)31-s + ⋯
L(s)  = 1  + (0.946 − 0.546i)3-s + (0.819 − 1.41i)5-s + (−0.108 − 0.994i)7-s + (0.0977 − 0.169i)9-s + (1.36 − 0.786i)11-s + 1.08i·13-s − 1.79i·15-s + (−0.259 + 0.149i)17-s + (0.836 + 0.483i)19-s + (−0.646 − 0.881i)21-s + (−0.290 + 0.503i)23-s + (−0.842 − 1.46i)25-s + 0.879i·27-s − 0.893i·29-s + (0.755 + 1.30i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0150 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0150 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.0150 + 0.999i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ -0.0150 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.697591134\)
\(L(\frac12)\) \(\approx\) \(2.697591134\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.288 + 2.63i)T \)
41 \( 1 + (4.85 + 4.17i)T \)
good3 \( 1 + (-1.64 + 0.946i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.83 + 3.17i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-4.51 + 2.60i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.89iT - 13T^{2} \)
17 \( 1 + (1.07 - 0.618i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.64 - 2.10i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.39 - 2.41i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.81iT - 29T^{2} \)
31 \( 1 + (-4.20 - 7.29i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.952 - 1.64i)T + (-18.5 - 32.0i)T^{2} \)
43 \( 1 + 11.5T + 43T^{2} \)
47 \( 1 + (-1.91 - 1.10i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.05 + 1.18i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.14 + 10.6i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.31 - 2.27i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.49 - 2.01i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 11.2iT - 71T^{2} \)
73 \( 1 + (-7.92 - 13.7i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.10 - 2.94i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 17.1T + 83T^{2} \)
89 \( 1 + (-1.77 - 1.02i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 2.27iT - 97T^{2} \)
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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

−9.475624461156282323349376983614, −8.633396960542242996920221077306, −8.303847962952888526939983153868, −7.10352396682353257730890729720, −6.38876310656560574593462758178, −5.28242160939951461134358523429, −4.25920127796640129531903068838, −3.37781494764653586466018724030, −1.76583604629704013424361122552, −1.20302911717930361279937461383, 1.98300449385845832923684119635, 2.89751633057780966834050035688, 3.43824958631947157247532366706, 4.79107179545010965795428052647, 6.01201690086920831796514368095, 6.56037855115360935990166290983, 7.53607160990538041395805892087, 8.612114959416491227041460057862, 9.352044387589978524811403324593, 9.826397902235295115234543768462

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