Properties

Label 2-1148-41.9-c1-0-6
Degree $2$
Conductor $1148$
Sign $0.989 + 0.144i$
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  + (−0.350 − 0.350i)3-s + 3.08i·5-s + (−0.707 − 0.707i)7-s − 2.75i·9-s + (1.14 + 1.14i)11-s + (−2.88 − 2.88i)13-s + (1.08 − 1.08i)15-s + (3.58 − 3.58i)17-s + (2.29 − 2.29i)19-s + 0.495i·21-s + 3.46·23-s − 4.51·25-s + (−2.01 + 2.01i)27-s + (4.60 + 4.60i)29-s + 2.08·31-s + ⋯
L(s)  = 1  + (−0.202 − 0.202i)3-s + 1.37i·5-s + (−0.267 − 0.267i)7-s − 0.917i·9-s + (0.344 + 0.344i)11-s + (−0.799 − 0.799i)13-s + (0.279 − 0.279i)15-s + (0.869 − 0.869i)17-s + (0.526 − 0.526i)19-s + 0.108i·21-s + 0.723·23-s − 0.902·25-s + (−0.388 + 0.388i)27-s + (0.855 + 0.855i)29-s + 0.374·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.144i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 + 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.989 + 0.144i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ 0.989 + 0.144i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.481845347\)
\(L(\frac12)\) \(\approx\) \(1.481845347\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 + 0.707i)T \)
41 \( 1 + (-6.11 + 1.90i)T \)
good3 \( 1 + (0.350 + 0.350i)T + 3iT^{2} \)
5 \( 1 - 3.08iT - 5T^{2} \)
11 \( 1 + (-1.14 - 1.14i)T + 11iT^{2} \)
13 \( 1 + (2.88 + 2.88i)T + 13iT^{2} \)
17 \( 1 + (-3.58 + 3.58i)T - 17iT^{2} \)
19 \( 1 + (-2.29 + 2.29i)T - 19iT^{2} \)
23 \( 1 - 3.46T + 23T^{2} \)
29 \( 1 + (-4.60 - 4.60i)T + 29iT^{2} \)
31 \( 1 - 2.08T + 31T^{2} \)
37 \( 1 - 6.46T + 37T^{2} \)
43 \( 1 - 6.95iT - 43T^{2} \)
47 \( 1 + (3.66 - 3.66i)T - 47iT^{2} \)
53 \( 1 + (-5.69 - 5.69i)T + 53iT^{2} \)
59 \( 1 - 2.42T + 59T^{2} \)
61 \( 1 + 8.24iT - 61T^{2} \)
67 \( 1 + (5.59 - 5.59i)T - 67iT^{2} \)
71 \( 1 + (3.14 + 3.14i)T + 71iT^{2} \)
73 \( 1 + 8.57iT - 73T^{2} \)
79 \( 1 + (-5.97 - 5.97i)T + 79iT^{2} \)
83 \( 1 - 13.2T + 83T^{2} \)
89 \( 1 + (3.25 + 3.25i)T + 89iT^{2} \)
97 \( 1 + (3.75 - 3.75i)T - 97iT^{2} \)
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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

−9.763353236707743114320991085976, −9.282972552065485792645820290381, −7.82164475495883265203467862577, −7.14811401163243429961649147981, −6.64778617400298141142950452109, −5.71698613560483897996589304464, −4.58413432564665620325565462883, −3.18700655625468249838749600001, −2.84109396342901266721127242815, −0.872932730752407173447328974103, 1.05989042450555251818808440981, 2.35370309808096976076237064151, 3.86659494852933506254092164430, 4.75246096651442764932470783789, 5.41480087048691430364516903357, 6.27956895342253662828820136099, 7.54625848430606562155884080851, 8.258818182517275701474151759463, 9.010711402451366627397441232983, 9.770000852009363715801324980440

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