Properties

Label 2-1183-7.2-c1-0-15
Degree $2$
Conductor $1183$
Sign $-0.806 + 0.591i$
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  + (−0.536 + 0.929i)2-s + (1.21 + 2.10i)3-s + (0.424 + 0.734i)4-s + (0.312 − 0.541i)5-s − 2.60·6-s + (−1.21 + 2.34i)7-s − 3.05·8-s + (−1.45 + 2.52i)9-s + (0.335 + 0.581i)10-s + (−0.354 − 0.613i)11-s + (−1.03 + 1.78i)12-s + (−1.53 − 2.39i)14-s + 1.52·15-s + (0.791 − 1.37i)16-s + (1.67 + 2.89i)17-s + (−1.56 − 2.70i)18-s + ⋯
L(s)  = 1  + (−0.379 + 0.657i)2-s + (0.701 + 1.21i)3-s + (0.212 + 0.367i)4-s + (0.139 − 0.242i)5-s − 1.06·6-s + (−0.459 + 0.888i)7-s − 1.08·8-s + (−0.485 + 0.840i)9-s + (0.106 + 0.183i)10-s + (−0.106 − 0.185i)11-s + (−0.297 + 0.515i)12-s + (−0.409 − 0.638i)14-s + 0.392·15-s + (0.197 − 0.342i)16-s + (0.405 + 0.702i)17-s + (−0.368 − 0.637i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.806 + 0.591i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.806 + 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.806 + 0.591i$
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: \((2,\ 1183,\ (\ :1/2),\ -0.806 + 0.591i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.370713847\)
\(L(\frac12)\) \(\approx\) \(1.370713847\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (1.21 - 2.34i)T \)
13 \( 1 \)
good2 \( 1 + (0.536 - 0.929i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-1.21 - 2.10i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.312 + 0.541i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.354 + 0.613i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.67 - 2.89i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.60 - 4.50i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.21 + 3.83i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.59T + 29T^{2} \)
31 \( 1 + (2.19 + 3.80i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.211 + 0.366i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 5.01T + 41T^{2} \)
43 \( 1 + 11.2T + 43T^{2} \)
47 \( 1 + (-4.03 + 6.99i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.348 - 0.603i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.93 - 8.54i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.34 - 4.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.21 - 9.02i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 14.0T + 71T^{2} \)
73 \( 1 + (2.54 + 4.40i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.95 + 3.38i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 10.2T + 83T^{2} \)
89 \( 1 + (6.68 - 11.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 0.202T + 97T^{2} \)
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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

−9.926001272035620849457228051975, −9.243314649564126795837946126607, −8.639063038499885817954585307812, −8.166181123021301344506451712638, −7.04166643038808900723889206172, −6.00760897294643508762878858614, −5.34257279655096965715268436053, −3.96919992984607356846256494890, −3.31901952585786618199264026565, −2.27787710121572428062245311322, 0.58395785825580217037634592343, 1.69140306602728976587835404955, 2.64948416622034324885678702805, 3.43318172584713187567205568933, 5.00055960210208333130245129641, 6.28605663872859585321064239167, 6.94276441925515355399694197348, 7.48297931543469837779008760285, 8.542411310023314626932377539686, 9.386421677089087563759885142651

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