Properties

Label 2-1183-13.12-c1-0-57
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  − 0.710i·2-s + 2.40·3-s + 1.49·4-s − 1.28i·5-s − 1.71i·6-s + i·7-s − 2.48i·8-s + 2.79·9-s − 0.916·10-s + 2.40i·11-s + 3.60·12-s + 0.710·14-s − 3.10i·15-s + 1.22·16-s − 3.90·17-s − 1.98i·18-s + ⋯
L(s)  = 1  − 0.502i·2-s + 1.39·3-s + 0.747·4-s − 0.576i·5-s − 0.698i·6-s + 0.377i·7-s − 0.877i·8-s + 0.932·9-s − 0.289·10-s + 0.726i·11-s + 1.03·12-s + 0.189·14-s − 0.801i·15-s + 0.306·16-s − 0.946·17-s − 0.468i·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

\(L(1)\) \(\approx\) \(3.203627540\)
\(L(\frac12)\) \(\approx\) \(3.203627540\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.710iT - 2T^{2} \)
3 \( 1 - 2.40T + 3T^{2} \)
5 \( 1 + 1.28iT - 5T^{2} \)
11 \( 1 - 2.40iT - 11T^{2} \)
17 \( 1 + 3.90T + 17T^{2} \)
19 \( 1 + 5.89iT - 19T^{2} \)
23 \( 1 - 6.32T + 23T^{2} \)
29 \( 1 - 5.61T + 29T^{2} \)
31 \( 1 - 2.20iT - 31T^{2} \)
37 \( 1 - 5.11iT - 37T^{2} \)
41 \( 1 + 7.78iT - 41T^{2} \)
43 \( 1 + 0.289T + 43T^{2} \)
47 \( 1 + 1.27iT - 47T^{2} \)
53 \( 1 + 13.6T + 53T^{2} \)
59 \( 1 - 4.03iT - 59T^{2} \)
61 \( 1 + 4.60T + 61T^{2} \)
67 \( 1 - 7.57iT - 67T^{2} \)
71 \( 1 - 7.22iT - 71T^{2} \)
73 \( 1 - 15.0iT - 73T^{2} \)
79 \( 1 + 9.30T + 79T^{2} \)
83 \( 1 + 1.36iT - 83T^{2} \)
89 \( 1 - 0.899iT - 89T^{2} \)
97 \( 1 - 15.6iT - 97T^{2} \)
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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

−9.407387237018452574772495359400, −8.938905077362198583654801505225, −8.234879034799678421009227633621, −7.12603914947670637176300413813, −6.68092967243334126084322667488, −5.12347021427949356443999595522, −4.23372437368351428300852439049, −2.94126698529732879537382977210, −2.52610896471531388508723719255, −1.34833379215562490137303545141, 1.69447839823033993183584966495, 2.86690805806240668517075588308, 3.32410106011350187701140812195, 4.65464440274907922988346947922, 6.02846620498842586020145066271, 6.69262135121475911923314172343, 7.54776564646867351967360644132, 8.142417933836834574792765759496, 8.833487791035478031632696224334, 9.746096004740673668078135790748

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