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Properties

Label 2-1183-7.4-c1-0-26
Degree $2$
Conductor $1183$
Sign $0.820 + 0.571i$
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  + (−1.37 − 2.37i)2-s + (−1.20 + 2.09i)3-s + (−2.76 + 4.78i)4-s + (−0.312 − 0.540i)5-s + 6.61·6-s + (1.27 + 2.31i)7-s + 9.65·8-s + (−1.41 − 2.44i)9-s + (−0.856 + 1.48i)10-s + (1.23 − 2.13i)11-s + (−6.66 − 11.5i)12-s + (3.76 − 6.20i)14-s + 1.50·15-s + (−7.72 − 13.3i)16-s + (0.00903 − 0.0156i)17-s + (−3.87 + 6.70i)18-s + ⋯
L(s)  = 1  + (−0.969 − 1.67i)2-s + (−0.696 + 1.20i)3-s + (−1.38 + 2.39i)4-s + (−0.139 − 0.241i)5-s + 2.70·6-s + (0.480 + 0.876i)7-s + 3.41·8-s + (−0.470 − 0.815i)9-s + (−0.270 + 0.468i)10-s + (0.371 − 0.644i)11-s + (−1.92 − 3.33i)12-s + (1.00 − 1.65i)14-s + 0.389·15-s + (−1.93 − 3.34i)16-s + (0.00219 − 0.00379i)17-s + (−0.913 + 1.58i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.820 + 0.571i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.820 + 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.820 + 0.571i$
Analytic conductor: \(9.44630\)
Root analytic conductor: \(3.07348\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (508, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1183,\ (\ :1/2),\ 0.820 + 0.571i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6801412865\)
\(L(\frac12)\) \(\approx\) \(0.6801412865\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-1.27 - 2.31i)T \)
13 \( 1 \)
good2 \( 1 + (1.37 + 2.37i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (1.20 - 2.09i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.312 + 0.540i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.23 + 2.13i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.00903 + 0.0156i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.67 + 4.62i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.66 - 2.87i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 4.10T + 29T^{2} \)
31 \( 1 + (-2.58 + 4.47i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.94 - 3.37i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 9.60T + 41T^{2} \)
43 \( 1 - 0.158T + 43T^{2} \)
47 \( 1 + (0.732 + 1.26i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.41 - 2.44i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.25 - 7.37i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.86 + 8.42i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.0205 + 0.0356i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.95T + 71T^{2} \)
73 \( 1 + (-6.26 + 10.8i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.40 - 7.63i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 13.2T + 83T^{2} \)
89 \( 1 + (1.30 + 2.25i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 5.04T + 97T^{2} \)
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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

−9.718384755457385516142489427618, −9.167032829197165878264855854619, −8.594777815213715514711754065846, −7.76515056695074739809702238135, −6.15243432904348798563908838857, −4.82895617681598592296049748501, −4.42618281933357889935496427415, −3.29153714737358098810114837863, −2.31452874804411596166600730957, −0.76222203201200296542784882874, 0.77895949738542816012124188879, 1.67744385079222792761153488574, 4.24193461416245516888720470193, 5.14203738541288703153415147904, 6.19577450471968034717094888895, 6.66162627380980647204796775395, 7.37203887138811227163987373744, 7.81379453557553181412868332564, 8.656352347978506013994088798290, 9.657130792229731585496723679786

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