Properties

Label 2-1183-13.12-c1-0-24
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  + 2.61i·2-s − 2.61·3-s − 4.85·4-s + 2.61i·5-s − 6.85i·6-s i·7-s − 7.47i·8-s + 3.85·9-s − 6.85·10-s − 1.85i·11-s + 12.7·12-s + 2.61·14-s − 6.85i·15-s + 9.85·16-s + 1.47·17-s + 10.0i·18-s + ⋯
L(s)  = 1  + 1.85i·2-s − 1.51·3-s − 2.42·4-s + 1.17i·5-s − 2.79i·6-s − 0.377i·7-s − 2.64i·8-s + 1.28·9-s − 2.16·10-s − 0.559i·11-s + 3.66·12-s + 0.699·14-s − 1.76i·15-s + 2.46·16-s + 0.357·17-s + 2.37i·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

\(L(1)\) \(\approx\) \(0.6851454920\)
\(L(\frac12)\) \(\approx\) \(0.6851454920\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.61iT - 2T^{2} \)
3 \( 1 + 2.61T + 3T^{2} \)
5 \( 1 - 2.61iT - 5T^{2} \)
11 \( 1 + 1.85iT - 11T^{2} \)
17 \( 1 - 1.47T + 17T^{2} \)
19 \( 1 + 1.85iT - 19T^{2} \)
23 \( 1 - 4.47T + 23T^{2} \)
29 \( 1 - 7.09T + 29T^{2} \)
31 \( 1 + 4.70iT - 31T^{2} \)
37 \( 1 + 4iT - 37T^{2} \)
41 \( 1 - 0.763iT - 41T^{2} \)
43 \( 1 + 12.5T + 43T^{2} \)
47 \( 1 - 2.23iT - 47T^{2} \)
53 \( 1 - 3.76T + 53T^{2} \)
59 \( 1 - 2.23iT - 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 + 12.7iT - 67T^{2} \)
71 \( 1 + 14.1iT - 71T^{2} \)
73 \( 1 - 2iT - 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 6.70iT - 83T^{2} \)
89 \( 1 + 4.90iT - 89T^{2} \)
97 \( 1 - 18.8iT - 97T^{2} \)
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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

−10.08556716707651377907038371688, −9.065701011558374629644367657984, −7.995967799851407947159370730143, −7.20655914977400169525899486687, −6.55284368690514802193420827397, −6.18511252449317536689309617781, −5.24670790399185567123742976358, −4.57600165197041968713615677515, −3.30271702574535043786347264086, −0.59295723061647359898440590104, 0.812109492845607523205066322376, 1.62644502057518434267803069052, 3.11190366982291820295760026603, 4.45123508975925128022489216101, 4.93235499036173288440157957498, 5.55804169668432054294894938583, 6.81742339835504787228716958904, 8.364188882902569677190534889660, 8.918754150586490194660209219330, 10.01334754546344607612683172777

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