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Properties

Label 2-1148-287.163-c1-0-2
Degree $2$
Conductor $1148$
Sign $-0.499 - 0.866i$
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  + (1.79 − 1.03i)3-s + (−0.904 + 1.56i)5-s + (−2.10 − 1.60i)7-s + (0.650 − 1.12i)9-s + (−2.79 + 1.61i)11-s + 6.58i·13-s + 3.75i·15-s + (−4.75 + 2.74i)17-s + (−4.51 − 2.60i)19-s + (−5.44 − 0.690i)21-s + (0.901 − 1.56i)23-s + (0.862 + 1.49i)25-s + 3.52i·27-s − 5.39i·29-s + (−2.38 − 4.12i)31-s + ⋯
L(s)  = 1  + (1.03 − 0.598i)3-s + (−0.404 + 0.700i)5-s + (−0.796 − 0.605i)7-s + (0.216 − 0.375i)9-s + (−0.842 + 0.486i)11-s + 1.82i·13-s + 0.969i·15-s + (−1.15 + 0.665i)17-s + (−1.03 − 0.598i)19-s + (−1.18 − 0.150i)21-s + (0.188 − 0.325i)23-s + (0.172 + 0.298i)25-s + 0.678i·27-s − 1.00i·29-s + (−0.428 − 0.741i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.499 - 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.499 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.499 - 0.866i$
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: \((2,\ 1148,\ (\ :1/2),\ -0.499 - 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8428594974\)
\(L(\frac12)\) \(\approx\) \(0.8428594974\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.10 + 1.60i)T \)
41 \( 1 + (-3.53 - 5.33i)T \)
good3 \( 1 + (-1.79 + 1.03i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.904 - 1.56i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.79 - 1.61i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 6.58iT - 13T^{2} \)
17 \( 1 + (4.75 - 2.74i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.51 + 2.60i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.901 + 1.56i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 5.39iT - 29T^{2} \)
31 \( 1 + (2.38 + 4.12i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.15 - 7.19i)T + (-18.5 - 32.0i)T^{2} \)
43 \( 1 - 5.81T + 43T^{2} \)
47 \( 1 + (2.28 + 1.31i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-8.66 + 5.00i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.69 + 8.13i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.70 + 4.69i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (9.69 - 5.59i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.46iT - 71T^{2} \)
73 \( 1 + (-7.19 - 12.4i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.86 + 3.96i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 0.937T + 83T^{2} \)
89 \( 1 + (7.40 + 4.27i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 8.46iT - 97T^{2} \)
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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

−9.980960498863920715180691187877, −9.122402155298840976208815436751, −8.429096353072435802232315917495, −7.49064907788419296880282298169, −6.87428028317869317167347571787, −6.36890297191878325155843031551, −4.57837408173633299680083552560, −3.84529675154524189777146141150, −2.67897401830907208445312249225, −1.99985443132231340534661573056, 0.29168932574297931649021362739, 2.50526576741731768484940890471, 3.15714371190863221556890189752, 4.09704116945527202999179841124, 5.21867511821783960867101152061, 5.94836231681967305339772730508, 7.28952491429078612233923399553, 8.197602006824960551077679638806, 8.839037833801755670292404270086, 9.157733835833205510308299451383

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