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Properties

Label 2-1148-287.163-c1-0-14
Degree $2$
Conductor $1148$
Sign $0.994 + 0.106i$
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  + (−1.59 + 0.920i)3-s + (1.70 − 2.95i)5-s + (2.57 + 0.601i)7-s + (0.195 − 0.339i)9-s + (4.43 − 2.55i)11-s + 4.75i·13-s + 6.28i·15-s + (−1.32 + 0.765i)17-s + (0.164 + 0.0948i)19-s + (−4.66 + 1.41i)21-s + (0.638 − 1.10i)23-s + (−3.31 − 5.74i)25-s − 4.80i·27-s + 6.12i·29-s + (−3.46 − 5.99i)31-s + ⋯
L(s)  = 1  + (−0.920 + 0.531i)3-s + (0.762 − 1.32i)5-s + (0.973 + 0.227i)7-s + (0.0653 − 0.113i)9-s + (1.33 − 0.771i)11-s + 1.32i·13-s + 1.62i·15-s + (−0.321 + 0.185i)17-s + (0.0376 + 0.0217i)19-s + (−1.01 + 0.308i)21-s + (0.133 − 0.230i)23-s + (−0.663 − 1.14i)25-s − 0.924i·27-s + 1.13i·29-s + (−0.621 − 1.07i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.106i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.106i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.994 + 0.106i$
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: \((2,\ 1148,\ (\ :1/2),\ 0.994 + 0.106i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.617386107\)
\(L(\frac12)\) \(\approx\) \(1.617386107\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.57 - 0.601i)T \)
41 \( 1 + (-5.72 - 2.86i)T \)
good3 \( 1 + (1.59 - 0.920i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.70 + 2.95i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-4.43 + 2.55i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 4.75iT - 13T^{2} \)
17 \( 1 + (1.32 - 0.765i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.164 - 0.0948i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.638 + 1.10i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.12iT - 29T^{2} \)
31 \( 1 + (3.46 + 5.99i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.21 + 9.03i)T + (-18.5 - 32.0i)T^{2} \)
43 \( 1 + 0.258T + 43T^{2} \)
47 \( 1 + (-2.76 - 1.59i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.49 + 3.17i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.59 - 9.68i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.82 - 11.8i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.11 - 0.643i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.74iT - 71T^{2} \)
73 \( 1 + (-1.22 - 2.11i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.75 + 3.89i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.45T + 83T^{2} \)
89 \( 1 + (-0.445 - 0.257i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 13.9iT - 97T^{2} \)
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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

−9.604430601953990482644599211385, −8.964214321230728003219904689844, −8.574258340524036985470798673840, −7.21283696864287700439731771967, −5.91009336085433435922360255478, −5.70241834624005894615928797957, −4.51826858521240467137823704303, −4.21727803052442701836797068772, −2.06401566022908151049713654400, −1.05267642709282140709763832666, 1.13667810648666343011174103646, 2.29067178685223435418961866503, 3.55419481789707136075698666370, 4.85879942556292826886075438226, 5.79412760015867858277219777344, 6.49255603611702063323518574439, 7.09520555997995166748670485000, 7.898255508233171554901903344595, 9.146979317101811190553167689665, 10.02572427898739704781465591725

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