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Properties

Label 2-1150-5.4-c1-0-8
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 + 3.11i·3-s − 4-s − 3.11·6-s + 4.50i·7-s − i·8-s − 6.72·9-s + 4.33·11-s − 3.11i·12-s + 3.72i·13-s − 4.50·14-s + 16-s + 1.11i·17-s − 6.72i·18-s − 4.50·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.80i·3-s − 0.5·4-s − 1.27·6-s + 1.70i·7-s − 0.353i·8-s − 2.24·9-s + 1.30·11-s − 0.900i·12-s + 1.03i·13-s − 1.20·14-s + 0.250·16-s + 0.271i·17-s − 1.58i·18-s − 1.03·19-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

\(L(1)\) \(\approx\) \(1.400222910\)
\(L(\frac12)\) \(\approx\) \(1.400222910\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.11iT - 3T^{2} \)
7 \( 1 - 4.50iT - 7T^{2} \)
11 \( 1 - 4.33T + 11T^{2} \)
13 \( 1 - 3.72iT - 13T^{2} \)
17 \( 1 - 1.11iT - 17T^{2} \)
19 \( 1 + 4.50T + 19T^{2} \)
29 \( 1 - 8.23T + 29T^{2} \)
31 \( 1 - 1.72T + 31T^{2} \)
37 \( 1 + 0.781iT - 37T^{2} \)
41 \( 1 - 3.90T + 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 + 11.4iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 - 2.23T + 59T^{2} \)
61 \( 1 - 3.55T + 61T^{2} \)
67 \( 1 - 2.43iT - 67T^{2} \)
71 \( 1 - 7.11T + 71T^{2} \)
73 \( 1 - 9.45iT - 73T^{2} \)
79 \( 1 - 14.9T + 79T^{2} \)
83 \( 1 + 2.78iT - 83T^{2} \)
89 \( 1 - 7.69T + 89T^{2} \)
97 \( 1 + 0.642iT - 97T^{2} \)
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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

−10.02933685703315855552205359540, −9.373540782287521625820071885044, −8.694567216881749317976809887821, −8.531121923300378097911099464731, −6.69595152108026386252218999609, −6.04019373163019993926386700090, −5.21094734394524584836971853641, −4.38582424447294994261595384175, −3.67210698886093732212753899882, −2.33734163032712120420745458619, 0.69254865095357059177978443704, 1.29316392511902349318088390131, 2.62031034995037913229378134318, 3.70426286237112574043637927993, 4.76459591049384446159368323857, 6.35064216412340256307899954303, 6.62480499318116692063902711901, 7.73375514506084037334009387410, 8.131098361610426389687239049113, 9.211290600947580527114849636775

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