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Properties

Label 2-1183-13.12-c1-0-14
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  − 0.842i·2-s + 0.161·3-s + 1.29·4-s + 3.72i·5-s − 0.136i·6-s + i·7-s − 2.77i·8-s − 2.97·9-s + 3.14·10-s + 3.51i·11-s + 0.208·12-s + 0.842·14-s + 0.603i·15-s + 0.244·16-s − 7.43·17-s + 2.50i·18-s + ⋯
L(s)  = 1  − 0.595i·2-s + 0.0935·3-s + 0.645·4-s + 1.66i·5-s − 0.0557i·6-s + 0.377i·7-s − 0.980i·8-s − 0.991·9-s + 0.993·10-s + 1.05i·11-s + 0.0603·12-s + 0.225·14-s + 0.155i·15-s + 0.0611·16-s − 1.80·17-s + 0.590i·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

\(L(1)\) \(\approx\) \(1.414213909\)
\(L(\frac12)\) \(\approx\) \(1.414213909\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.842iT - 2T^{2} \)
3 \( 1 - 0.161T + 3T^{2} \)
5 \( 1 - 3.72iT - 5T^{2} \)
11 \( 1 - 3.51iT - 11T^{2} \)
17 \( 1 + 7.43T + 17T^{2} \)
19 \( 1 - 2.67iT - 19T^{2} \)
23 \( 1 - 2.49T + 23T^{2} \)
29 \( 1 - 7.30T + 29T^{2} \)
31 \( 1 + 2.54iT - 31T^{2} \)
37 \( 1 + 2.17iT - 37T^{2} \)
41 \( 1 - 8.40iT - 41T^{2} \)
43 \( 1 + 11.8T + 43T^{2} \)
47 \( 1 - 9.40iT - 47T^{2} \)
53 \( 1 + 5.84T + 53T^{2} \)
59 \( 1 - 4.25iT - 59T^{2} \)
61 \( 1 - 5.07T + 61T^{2} \)
67 \( 1 - 1.29iT - 67T^{2} \)
71 \( 1 + 1.30iT - 71T^{2} \)
73 \( 1 + 4.31iT - 73T^{2} \)
79 \( 1 - 14.2T + 79T^{2} \)
83 \( 1 - 9.81iT - 83T^{2} \)
89 \( 1 + 4.77iT - 89T^{2} \)
97 \( 1 - 5.92iT - 97T^{2} \)
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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

−10.15185354368700676365377280738, −9.454230808038310242993609175354, −8.293908491404720488891405456524, −7.34695549551793228187108416449, −6.53741788293054419461918166908, −6.20391839004865826831114262690, −4.65682887318739910285630185201, −3.38425256937593255420486037297, −2.65871926465782632448615978988, −2.02425644168679282048610384915, 0.54090813196160237475090124819, 2.05408225867456509122152001247, 3.30455393121990239926876115892, 4.72697196942929869356981326600, 5.26595987593613138608590427380, 6.25429972250205306449057218497, 6.94740623620230298936700112662, 8.274654526821271802873100985522, 8.523368492022706275722668223887, 9.095473251873167845223273787271

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