Properties

Label 2-1150-5.4-c1-0-7
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 − 1.61i·3-s − 4-s + 1.61·6-s − 0.618i·7-s i·8-s + 0.381·9-s − 2.85·11-s + 1.61i·12-s + 7.09i·13-s + 0.618·14-s + 16-s + 6.09i·17-s + 0.381i·18-s − 1.85·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.934i·3-s − 0.5·4-s + 0.660·6-s − 0.233i·7-s − 0.353i·8-s + 0.127·9-s − 0.860·11-s + 0.467i·12-s + 1.96i·13-s + 0.165·14-s + 0.250·16-s + 1.47i·17-s + 0.0900i·18-s − 0.425·19-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

\(L(1)\) \(\approx\) \(1.383967376\)
\(L(\frac12)\) \(\approx\) \(1.383967376\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.61iT - 3T^{2} \)
7 \( 1 + 0.618iT - 7T^{2} \)
11 \( 1 + 2.85T + 11T^{2} \)
13 \( 1 - 7.09iT - 13T^{2} \)
17 \( 1 - 6.09iT - 17T^{2} \)
19 \( 1 + 1.85T + 19T^{2} \)
29 \( 1 - 9.23T + 29T^{2} \)
31 \( 1 - 9.09T + 31T^{2} \)
37 \( 1 - 6.47iT - 37T^{2} \)
41 \( 1 - 3.32T + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 3.70iT - 47T^{2} \)
53 \( 1 + 0.472iT - 53T^{2} \)
59 \( 1 + 1.70T + 59T^{2} \)
61 \( 1 + 9.32T + 61T^{2} \)
67 \( 1 - 14.4iT - 67T^{2} \)
71 \( 1 + 4.09T + 71T^{2} \)
73 \( 1 + 3.23iT - 73T^{2} \)
79 \( 1 + 1.52T + 79T^{2} \)
83 \( 1 - 6.94iT - 83T^{2} \)
89 \( 1 - 10.4T + 89T^{2} \)
97 \( 1 - 12.3iT - 97T^{2} \)
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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

−9.972340179731359512701532667216, −8.749843468703165277228847752102, −8.220706016084148555060942103289, −7.37818256170665495953152424854, −6.51549234017036584577673124209, −6.24104245286751234453626169874, −4.73116225726202409921128655238, −4.11714581126645284852176468989, −2.45356271276766365201936024083, −1.27142648150277231074973219052, 0.68572492112958794239577092208, 2.64640992687226977352008242593, 3.18467390660368093446184141479, 4.52985477354801688432311110144, 5.04637358628118877702860369581, 5.97143921702078167435448424108, 7.42386410101485873984954901214, 8.166394903683140404070375762343, 9.086892404835865286996746173776, 9.908218696093931152473498273671

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