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Properties

Label 2-1183-13.12-c1-0-64
Degree $2$
Conductor $1183$
Sign $-0.554 + 0.832i$
Analytic cond. $9.44630$
Root an. cond. $3.07348$
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  − 1.73i·2-s + 2.73·3-s − 0.999·4-s − 1.73i·5-s − 4.73i·6-s + i·7-s − 1.73i·8-s + 4.46·9-s − 2.99·10-s + 1.26i·11-s − 2.73·12-s + 1.73·14-s − 4.73i·15-s − 5·16-s + 7.73·17-s − 7.73i·18-s + ⋯
L(s)  = 1  − 1.22i·2-s + 1.57·3-s − 0.499·4-s − 0.774i·5-s − 1.93i·6-s + 0.377i·7-s − 0.612i·8-s + 1.48·9-s − 0.948·10-s + 0.382i·11-s − 0.788·12-s + 0.462·14-s − 1.22i·15-s − 1.25·16-s + 1.87·17-s − 1.82i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.554 + 0.832i$
Analytic conductor: \(9.44630\)
Root analytic conductor: \(3.07348\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1183,\ (\ :1/2),\ -0.554 + 0.832i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.029092134\)
\(L(\frac12)\) \(\approx\) \(3.029092134\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 - iT \)
13 \( 1 \)
good2 \( 1 + 1.73iT - 2T^{2} \)
3 \( 1 - 2.73T + 3T^{2} \)
5 \( 1 + 1.73iT - 5T^{2} \)
11 \( 1 - 1.26iT - 11T^{2} \)
17 \( 1 - 7.73T + 17T^{2} \)
19 \( 1 + 2iT - 19T^{2} \)
23 \( 1 + 4.73T + 23T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 + 4.19iT - 31T^{2} \)
37 \( 1 + 7iT - 37T^{2} \)
41 \( 1 - 5.19iT - 41T^{2} \)
43 \( 1 - 0.196T + 43T^{2} \)
47 \( 1 - 12.9iT - 47T^{2} \)
53 \( 1 + 9.92T + 53T^{2} \)
59 \( 1 - 7.26iT - 59T^{2} \)
61 \( 1 + 4.80T + 61T^{2} \)
67 \( 1 - 6.19iT - 67T^{2} \)
71 \( 1 + 6iT - 71T^{2} \)
73 \( 1 + 3.19iT - 73T^{2} \)
79 \( 1 - 16.1T + 79T^{2} \)
83 \( 1 - 2.19iT - 83T^{2} \)
89 \( 1 - 12.9iT - 89T^{2} \)
97 \( 1 + 6.39iT - 97T^{2} \)
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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

−9.460910000969207713050154474709, −9.032563918349042475060470476983, −7.964313849954526032080257996232, −7.50219690090582809594128673088, −6.04899764144146591536673308065, −4.74310268871806509305129714468, −3.81365291971494718725473275225, −3.03518261262074460495484556805, −2.17513959996215953221793590737, −1.20829327941474086115363433689, 1.86042391222461666486964858666, 3.09851232378160056578501660679, 3.67340926815685343710078213848, 5.09243028882945964949667444822, 6.11901300558811530125916415329, 6.97236013058090716514445831135, 7.71356295466763567578111935076, 8.096939424870252097694830415803, 8.887968205226666624580668405734, 9.877447206307190006083663013811

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