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Properties

Label 2-1183-13.12-c1-0-39
Degree $2$
Conductor $1183$
Sign $0.960 + 0.277i$
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  + 0.120i·2-s − 0.582·3-s + 1.98·4-s − 1.68i·5-s − 0.0700i·6-s + i·7-s + 0.479i·8-s − 2.66·9-s + 0.203·10-s + 0.364i·11-s − 1.15·12-s − 0.120·14-s + 0.983i·15-s + 3.91·16-s + 3.18·17-s − 0.320i·18-s + ⋯
L(s)  = 1  + 0.0851i·2-s − 0.336·3-s + 0.992·4-s − 0.754i·5-s − 0.0286i·6-s + 0.377i·7-s + 0.169i·8-s − 0.886·9-s + 0.0642·10-s + 0.109i·11-s − 0.333·12-s − 0.0321·14-s + 0.253i·15-s + 0.978·16-s + 0.772·17-s − 0.0754i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.277i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.960 + 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.960 + 0.277i$
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: \((2,\ 1183,\ (\ :1/2),\ 0.960 + 0.277i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.837603051\)
\(L(\frac12)\) \(\approx\) \(1.837603051\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.120iT - 2T^{2} \)
3 \( 1 + 0.582T + 3T^{2} \)
5 \( 1 + 1.68iT - 5T^{2} \)
11 \( 1 - 0.364iT - 11T^{2} \)
17 \( 1 - 3.18T + 17T^{2} \)
19 \( 1 - 1.44iT - 19T^{2} \)
23 \( 1 - 5.08T + 23T^{2} \)
29 \( 1 - 8.19T + 29T^{2} \)
31 \( 1 + 4.69iT - 31T^{2} \)
37 \( 1 + 6.31iT - 37T^{2} \)
41 \( 1 - 5.82iT - 41T^{2} \)
43 \( 1 - 0.773T + 43T^{2} \)
47 \( 1 + 12.7iT - 47T^{2} \)
53 \( 1 - 1.37T + 53T^{2} \)
59 \( 1 + 9.36iT - 59T^{2} \)
61 \( 1 + 9.02T + 61T^{2} \)
67 \( 1 - 13.4iT - 67T^{2} \)
71 \( 1 + 7.08iT - 71T^{2} \)
73 \( 1 - 2.16iT - 73T^{2} \)
79 \( 1 + 6.88T + 79T^{2} \)
83 \( 1 - 0.567iT - 83T^{2} \)
89 \( 1 - 1.13iT - 89T^{2} \)
97 \( 1 - 7.92iT - 97T^{2} \)
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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

−9.774131717797751563754107048066, −8.746361180755440389346834748866, −8.178203416539504781289891324136, −7.19645693022430073861297131431, −6.27789705992377044622635583393, −5.56689425571200498886681555260, −4.83896851688717241833468010240, −3.34847680222986481904097093270, −2.40601194456677691036876494188, −1.00975201449042475412682878366, 1.16172641029127390943573080499, 2.84331039404909633437384373153, 3.13393695521962893221540532767, 4.73820933565547498384804280219, 5.78114583223169512480793381282, 6.54979908673557340432226828131, 7.13385620002675132085117365979, 8.024891698550563856393621085725, 8.966514579290536385783102659768, 10.16348369097686102372393135047

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