Properties

Label 2-1150-5.4-c1-0-17
Degree $2$
Conductor $1150$
Sign $0.447 - 0.894i$
Analytic cond. $9.18279$
Root an. cond. $3.03031$
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  + i·2-s + 0.618i·3-s − 4-s − 0.618·6-s + 1.61i·7-s i·8-s + 2.61·9-s + 3.85·11-s − 0.618i·12-s − 4.09i·13-s − 1.61·14-s + 16-s − 5.09i·17-s + 2.61i·18-s + 4.85·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.356i·3-s − 0.5·4-s − 0.252·6-s + 0.611i·7-s − 0.353i·8-s + 0.872·9-s + 1.16·11-s − 0.178i·12-s − 1.13i·13-s − 0.432·14-s + 0.250·16-s − 1.23i·17-s + 0.617i·18-s + 1.11·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1150\)    =    \(2 \cdot 5^{2} \cdot 23\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(9.18279\)
Root analytic conductor: \(3.03031\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1150} (599, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1150,\ (\ :1/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.813493148\)
\(L(\frac12)\) \(\approx\) \(1.813493148\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
5 \( 1 \)
23 \( 1 + iT \)
good3 \( 1 - 0.618iT - 3T^{2} \)
7 \( 1 - 1.61iT - 7T^{2} \)
11 \( 1 - 3.85T + 11T^{2} \)
13 \( 1 + 4.09iT - 13T^{2} \)
17 \( 1 + 5.09iT - 17T^{2} \)
19 \( 1 - 4.85T + 19T^{2} \)
29 \( 1 - 4.76T + 29T^{2} \)
31 \( 1 + 2.09T + 31T^{2} \)
37 \( 1 + 2.47iT - 37T^{2} \)
41 \( 1 + 12.3T + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 - 9.70iT - 47T^{2} \)
53 \( 1 - 8.47iT - 53T^{2} \)
59 \( 1 - 11.7T + 59T^{2} \)
61 \( 1 - 6.32T + 61T^{2} \)
67 \( 1 - 5.52iT - 67T^{2} \)
71 \( 1 - 7.09T + 71T^{2} \)
73 \( 1 - 1.23iT - 73T^{2} \)
79 \( 1 + 10.4T + 79T^{2} \)
83 \( 1 + 10.9iT - 83T^{2} \)
89 \( 1 - 1.52T + 89T^{2} \)
97 \( 1 - 14.6iT - 97T^{2} \)
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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

−9.713189419846584181181313621733, −9.185219560691512855975838922228, −8.311198866374228622852868008555, −7.33577273748477952740903663362, −6.73230416182722236229586333079, −5.62172846952858172143661652394, −4.96907361019676594790498805106, −3.94534862451535870191446265769, −2.89349603939038432008604778389, −1.09598191234769499830773332591, 1.16448812169687817509094237970, 1.92255093531253886581469112916, 3.63443890204363162497164048483, 4.08420315522076697231963854073, 5.19638848581168430759171761565, 6.62215926578494399074139728032, 6.95848288594634006033874891950, 8.143139290951214611019996904129, 8.943498963343155621504905396376, 9.891100287966834553906229568108

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