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Properties

Label 2-1183-13.12-c1-0-33
Degree $2$
Conductor $1183$
Sign $0.722 - 0.691i$
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  + 0.983i·2-s − 1.57·3-s + 1.03·4-s + 0.398i·5-s − 1.54i·6-s − i·7-s + 2.98i·8-s − 0.529·9-s − 0.391·10-s − 4.24i·11-s − 1.62·12-s + 0.983·14-s − 0.626i·15-s − 0.870·16-s + 5.10·17-s − 0.521i·18-s + ⋯
L(s)  = 1  + 0.695i·2-s − 0.907·3-s + 0.516·4-s + 0.178i·5-s − 0.631i·6-s − 0.377i·7-s + 1.05i·8-s − 0.176·9-s − 0.123·10-s − 1.27i·11-s − 0.468·12-s + 0.262·14-s − 0.161i·15-s − 0.217·16-s + 1.23·17-s − 0.122i·18-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.722 - 0.691i$
Analytic conductor: \(9.44630\)
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: \((2,\ 1183,\ (\ :1/2),\ 0.722 - 0.691i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.405090078\)
\(L(\frac12)\) \(\approx\) \(1.405090078\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.983iT - 2T^{2} \)
3 \( 1 + 1.57T + 3T^{2} \)
5 \( 1 - 0.398iT - 5T^{2} \)
11 \( 1 + 4.24iT - 11T^{2} \)
17 \( 1 - 5.10T + 17T^{2} \)
19 \( 1 - 2.12iT - 19T^{2} \)
23 \( 1 + 2.19T + 23T^{2} \)
29 \( 1 - 2.90T + 29T^{2} \)
31 \( 1 - 2.20iT - 31T^{2} \)
37 \( 1 + 11.4iT - 37T^{2} \)
41 \( 1 - 5.07iT - 41T^{2} \)
43 \( 1 - 0.328T + 43T^{2} \)
47 \( 1 + 6.62iT - 47T^{2} \)
53 \( 1 - 10.9T + 53T^{2} \)
59 \( 1 - 7.70iT - 59T^{2} \)
61 \( 1 - 14.6T + 61T^{2} \)
67 \( 1 - 1.22iT - 67T^{2} \)
71 \( 1 + 1.75iT - 71T^{2} \)
73 \( 1 - 6.11iT - 73T^{2} \)
79 \( 1 - 4.20T + 79T^{2} \)
83 \( 1 - 10.8iT - 83T^{2} \)
89 \( 1 - 15.6iT - 89T^{2} \)
97 \( 1 + 14.4iT - 97T^{2} \)
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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

−10.17138744518164738771931078305, −8.739762855480685996896571111563, −8.126923074275898832578832578846, −7.19722279143311849529118980008, −6.45974837293074423552739486973, −5.67430332179869265930853536124, −5.29101387175680103111173496930, −3.73803401316862678944346119095, −2.67073103845992483518000577731, −0.925403861123435368805931456420, 0.971640414833515175997645849942, 2.24814371900872787862372524870, 3.24706696582998438317687771043, 4.56341095170484072998254880994, 5.37331408022243021956384664190, 6.32950167391484361651545434521, 7.00398449171251019210494940680, 7.969570092436168440184363798515, 9.079098103754906849206359920579, 10.09381115150737767392685280736

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