Properties

Label 2-1183-7.2-c1-0-62
Degree $2$
Conductor $1183$
Sign $0.905 - 0.423i$
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  + (0.0904 − 0.156i)2-s + (0.913 + 1.58i)3-s + (0.983 + 1.70i)4-s + (1.34 − 2.32i)5-s + 0.330·6-s + (1.64 − 2.06i)7-s + 0.717·8-s + (−0.167 + 0.289i)9-s + (−0.242 − 0.420i)10-s + (1.34 + 2.33i)11-s + (−1.79 + 3.11i)12-s + (−0.174 − 0.445i)14-s + 4.90·15-s + (−1.90 + 3.29i)16-s + (−2.38 − 4.12i)17-s + (0.0302 + 0.0523i)18-s + ⋯
L(s)  = 1  + (0.0639 − 0.110i)2-s + (0.527 + 0.913i)3-s + (0.491 + 0.851i)4-s + (0.600 − 1.04i)5-s + 0.134·6-s + (0.623 − 0.781i)7-s + 0.253·8-s + (−0.0557 + 0.0965i)9-s + (−0.0768 − 0.133i)10-s + (0.406 + 0.703i)11-s + (−0.518 + 0.898i)12-s + (−0.0467 − 0.119i)14-s + 1.26·15-s + (−0.475 + 0.823i)16-s + (−0.577 − 1.00i)17-s + (0.00712 + 0.0123i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.905 - 0.423i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.905 - 0.423i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.905 - 0.423i$
Analytic conductor: \(9.44630\)
Root analytic conductor: \(3.07348\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (170, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1183,\ (\ :1/2),\ 0.905 - 0.423i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.809659130\)
\(L(\frac12)\) \(\approx\) \(2.809659130\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-1.64 + 2.06i)T \)
13 \( 1 \)
good2 \( 1 + (-0.0904 + 0.156i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-0.913 - 1.58i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.34 + 2.32i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.34 - 2.33i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.38 + 4.12i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.0942 + 0.163i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.19 - 3.80i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.08T + 29T^{2} \)
31 \( 1 + (1.84 + 3.20i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.97 + 6.88i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 5.42T + 41T^{2} \)
43 \( 1 + 8.01T + 43T^{2} \)
47 \( 1 + (0.924 - 1.60i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.53 - 6.12i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.79 - 6.57i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.205 + 0.356i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.70 - 9.87i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.34T + 71T^{2} \)
73 \( 1 + (7.10 + 12.3i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.55 - 7.89i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 16.5T + 83T^{2} \)
89 \( 1 + (2.94 - 5.10i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 0.451T + 97T^{2} \)
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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

−9.724374521102704244816961059846, −9.048616590184755062224774768192, −8.369346364613095433438536140099, −7.43762986819776154626682845850, −6.68099874757972685995518423784, −5.22800790699237360649578533645, −4.40940320510386147723774738934, −3.90169273624964345308057410696, −2.62229045021992933035157118523, −1.40533790449176884829192355619, 1.47948249336819215562165021614, 2.20479405700625794246600497273, 3.01499540173905132781381743265, 4.72899306771453813408606806514, 5.80195446953307060301720762403, 6.57724382340582473146587184801, 6.83991702178634563109554651672, 8.246075501517252300682414826985, 8.540667522196691697631382425397, 9.910989134797964988596125103371

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