Properties

Label 2-1183-13.12-c1-0-38
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  + 0.231i·2-s − 3.32·3-s + 1.94·4-s − 2.23i·5-s − 0.768i·6-s + i·7-s + 0.913i·8-s + 8.03·9-s + 0.516·10-s − 3.32i·11-s − 6.46·12-s − 0.231·14-s + 7.41i·15-s + 3.68·16-s + 1.37·17-s + 1.85i·18-s + ⋯
L(s)  = 1  + 0.163i·2-s − 1.91·3-s + 0.973·4-s − 0.997i·5-s − 0.313i·6-s + 0.377i·7-s + 0.322i·8-s + 2.67·9-s + 0.163·10-s − 1.00i·11-s − 1.86·12-s − 0.0618·14-s + 1.91i·15-s + 0.920·16-s + 0.333·17-s + 0.438i·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

\(L(1)\) \(\approx\) \(1.064164503\)
\(L(\frac12)\) \(\approx\) \(1.064164503\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.231iT - 2T^{2} \)
3 \( 1 + 3.32T + 3T^{2} \)
5 \( 1 + 2.23iT - 5T^{2} \)
11 \( 1 + 3.32iT - 11T^{2} \)
17 \( 1 - 1.37T + 17T^{2} \)
19 \( 1 - 3.23iT - 19T^{2} \)
23 \( 1 + 0.838T + 23T^{2} \)
29 \( 1 + 0.607T + 29T^{2} \)
31 \( 1 - 1.71iT - 31T^{2} \)
37 \( 1 + 1.55iT - 37T^{2} \)
41 \( 1 + 9.17iT - 41T^{2} \)
43 \( 1 + 1.23T + 43T^{2} \)
47 \( 1 - 1.62iT - 47T^{2} \)
53 \( 1 - 8.39T + 53T^{2} \)
59 \( 1 + 8.82iT - 59T^{2} \)
61 \( 1 - 5.46T + 61T^{2} \)
67 \( 1 + 10.1iT - 67T^{2} \)
71 \( 1 + 5.21iT - 71T^{2} \)
73 \( 1 - 3.96iT - 73T^{2} \)
79 \( 1 - 6.45T + 79T^{2} \)
83 \( 1 + 4.64iT - 83T^{2} \)
89 \( 1 + 9.12iT - 89T^{2} \)
97 \( 1 + 15.3iT - 97T^{2} \)
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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

−9.940249243371868044412791776439, −8.782330972081652586046860385206, −7.80885531878443427815425351200, −6.89760912430429960267016588950, −6.03361398760669114622706326985, −5.60223787372390552319039349980, −4.89726104643860147500911777199, −3.63482554156963940977545663370, −1.79193834449114359712912237937, −0.68851975095444636229562804839, 1.15174057452901347162689897001, 2.46557437504673626176629000262, 3.88349549384318340142674053311, 4.91825444010047641647545484934, 5.85236450901598381756347460780, 6.71252813142597171615164715306, 6.97718451559403144638552831303, 7.73413845839608585545245026945, 9.652245519636465125473269605903, 10.31167093434163032849650917489

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