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Properties

Label 2-1148-41.9-c1-0-20
Degree $2$
Conductor $1148$
Sign $-0.0613 - 0.998i$
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  + (−1.89 − 1.89i)3-s − 1.39i·5-s + (−0.707 − 0.707i)7-s + 4.19i·9-s + (−0.00674 − 0.00674i)11-s + (−3.61 − 3.61i)13-s + (−2.64 + 2.64i)15-s + (−2.98 + 2.98i)17-s + (3.03 − 3.03i)19-s + 2.68i·21-s − 6.42·23-s + 3.05·25-s + (2.25 − 2.25i)27-s + (6.04 + 6.04i)29-s − 9.06·31-s + ⋯
L(s)  = 1  + (−1.09 − 1.09i)3-s − 0.623i·5-s + (−0.267 − 0.267i)7-s + 1.39i·9-s + (−0.00203 − 0.00203i)11-s + (−1.00 − 1.00i)13-s + (−0.682 + 0.682i)15-s + (−0.724 + 0.724i)17-s + (0.695 − 0.695i)19-s + 0.585i·21-s − 1.34·23-s + 0.611·25-s + (0.434 − 0.434i)27-s + (1.12 + 1.12i)29-s − 1.62·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0613 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0613 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.0613 - 0.998i$
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: \((2,\ 1148,\ (\ :1/2),\ -0.0613 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.06509462155\)
\(L(\frac12)\) \(\approx\) \(0.06509462155\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.12 + 5.59i)T \)
good3 \( 1 + (1.89 + 1.89i)T + 3iT^{2} \)
5 \( 1 + 1.39iT - 5T^{2} \)
11 \( 1 + (0.00674 + 0.00674i)T + 11iT^{2} \)
13 \( 1 + (3.61 + 3.61i)T + 13iT^{2} \)
17 \( 1 + (2.98 - 2.98i)T - 17iT^{2} \)
19 \( 1 + (-3.03 + 3.03i)T - 19iT^{2} \)
23 \( 1 + 6.42T + 23T^{2} \)
29 \( 1 + (-6.04 - 6.04i)T + 29iT^{2} \)
31 \( 1 + 9.06T + 31T^{2} \)
37 \( 1 + 2.92T + 37T^{2} \)
43 \( 1 + 7.31iT - 43T^{2} \)
47 \( 1 + (-3.23 + 3.23i)T - 47iT^{2} \)
53 \( 1 + (-9.80 - 9.80i)T + 53iT^{2} \)
59 \( 1 - 8.32T + 59T^{2} \)
61 \( 1 - 12.1iT - 61T^{2} \)
67 \( 1 + (7.17 - 7.17i)T - 67iT^{2} \)
71 \( 1 + (0.0715 + 0.0715i)T + 71iT^{2} \)
73 \( 1 - 15.1iT - 73T^{2} \)
79 \( 1 + (-7.11 - 7.11i)T + 79iT^{2} \)
83 \( 1 - 0.354T + 83T^{2} \)
89 \( 1 + (2.66 + 2.66i)T + 89iT^{2} \)
97 \( 1 + (-3.41 + 3.41i)T - 97iT^{2} \)
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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

−9.078292288893822086634208978450, −8.283721083878605330235584479086, −7.13815195628189809234306148671, −6.96240396473621324369123170503, −5.58242512272425269798470496734, −5.32979015715303143833438062443, −4.03495138879725404973358311943, −2.48427860189493478347726913471, −1.17474010929646639308541780964, −0.03549696930490546388483595735, 2.25721455953354476365389326125, 3.54839142044700868842909816648, 4.51487357817508573919290705407, 5.20332789568186542316764095406, 6.20215570257290092039844592047, 6.80752454149936570815761296086, 7.86631656173805349003018714561, 9.196257836864679357287788475052, 9.741756768504994824126465636706, 10.33296729766860114542826796500

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