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Properties

Label 2-1183-91.90-c0-0-2
Degree $2$
Conductor $1183$
Sign $0.722 - 0.691i$
Analytic cond. $0.590393$
Root an. cond. $0.768370$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.445i·2-s + 0.801·4-s + i·7-s + 0.801i·8-s + 9-s − 1.80i·11-s − 0.445·14-s + 0.445·16-s + 0.445i·18-s + 0.801·22-s − 1.24·23-s − 25-s + 0.801i·28-s − 0.445·29-s + i·32-s + ⋯
L(s)  = 1  + 0.445i·2-s + 0.801·4-s + i·7-s + 0.801i·8-s + 9-s − 1.80i·11-s − 0.445·14-s + 0.445·16-s + 0.445i·18-s + 0.801·22-s − 1.24·23-s − 25-s + 0.801i·28-s − 0.445·29-s + i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.722 - 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.722 - 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.722 - 0.691i$
Analytic conductor: \(0.590393\)
Root analytic conductor: \(0.768370\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (1182, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1183,\ (\ :0),\ 0.722 - 0.691i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.345294276\)
\(L(\frac12)\) \(\approx\) \(1.345294276\)
\(L(1)\) 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 \( 1 - 0.445iT - T^{2} \)
3 \( 1 - T^{2} \)
5 \( 1 + T^{2} \)
11 \( 1 + 1.80iT - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + 1.24T + T^{2} \)
29 \( 1 + 0.445T + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - 1.24iT - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - 0.445T + T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - 1.24T + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + 1.24iT - T^{2} \)
71 \( 1 + 1.24iT - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + 1.80T + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + T^{2} \)
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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

−10.09676375656118201294554068623, −9.121719484398225494226167810666, −8.229377942458507153092479637109, −7.72310172682735922349114582605, −6.55093293142049851757150267198, −5.99429603769846629070375195729, −5.29861971252473954102078130117, −3.87392417620014023274084582909, −2.81655204007941368478519604304, −1.70246991785693939598601056484, 1.51167551224613997832990203857, 2.31269360316136767014671361751, 3.95186000240220855512910664642, 4.23946648585480425412748279776, 5.69634159195607649558376356692, 6.90585119695914785178361063045, 7.23773452861073839712127100414, 7.914303042561894941822798832704, 9.510006750215402807322884576075, 10.03290835587250833333416103605

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