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Properties

Label 2-1183-13.12-c1-0-65
Degree $2$
Conductor $1183$
Sign $0.0304 + 0.999i$
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  − 0.312i·2-s + 1.10·3-s + 1.90·4-s − 1.93i·5-s − 0.343i·6-s − i·7-s − 1.21i·8-s − 1.78·9-s − 0.603·10-s − 3.30i·11-s + 2.09·12-s − 0.312·14-s − 2.12i·15-s + 3.42·16-s + 2.68·17-s + 0.558i·18-s + ⋯
L(s)  = 1  − 0.220i·2-s + 0.635·3-s + 0.951·4-s − 0.865i·5-s − 0.140i·6-s − 0.377i·7-s − 0.430i·8-s − 0.596·9-s − 0.190·10-s − 0.996i·11-s + 0.604·12-s − 0.0834·14-s − 0.549i·15-s + 0.856·16-s + 0.650·17-s + 0.131i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0304 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0304 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.0304 + 0.999i$
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: \((2,\ 1183,\ (\ :1/2),\ 0.0304 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.376409997\)
\(L(\frac12)\) \(\approx\) \(2.376409997\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.312iT - 2T^{2} \)
3 \( 1 - 1.10T + 3T^{2} \)
5 \( 1 + 1.93iT - 5T^{2} \)
11 \( 1 + 3.30iT - 11T^{2} \)
17 \( 1 - 2.68T + 17T^{2} \)
19 \( 1 - 3.97iT - 19T^{2} \)
23 \( 1 - 0.249T + 23T^{2} \)
29 \( 1 + 6.85T + 29T^{2} \)
31 \( 1 + 8.16iT - 31T^{2} \)
37 \( 1 + 1.51iT - 37T^{2} \)
41 \( 1 + 1.36iT - 41T^{2} \)
43 \( 1 - 10.5T + 43T^{2} \)
47 \( 1 + 2.89iT - 47T^{2} \)
53 \( 1 + 11.3T + 53T^{2} \)
59 \( 1 - 13.9iT - 59T^{2} \)
61 \( 1 - 13.7T + 61T^{2} \)
67 \( 1 + 2.82iT - 67T^{2} \)
71 \( 1 - 9.17iT - 71T^{2} \)
73 \( 1 - 8.47iT - 73T^{2} \)
79 \( 1 + 8.68T + 79T^{2} \)
83 \( 1 + 8.89iT - 83T^{2} \)
89 \( 1 + 7.50iT - 89T^{2} \)
97 \( 1 - 8.70iT - 97T^{2} \)
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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

−9.500702537096670502308150481492, −8.708821381470100309916834504157, −7.948190492555078229229911650492, −7.35327218122893196590918276071, −5.97687525039013198731299732297, −5.59079438078197129740383454931, −4.02198033593075475037762477884, −3.26794937540369215777416311794, −2.20269653993229143718713857905, −0.931012881130360279583159909137, 1.89229264687998400835839577832, 2.76373130300890374740397314881, 3.40745647697277464568656553201, 4.97211233935055229995278829899, 5.93162011960262830896493824947, 6.82025639970473195859193302388, 7.40427919781292007620304264750, 8.172889323942056735715672348156, 9.147974791538018276480692574902, 9.910146222982363226242615174102

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