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Properties

Label 2-1183-7.2-c1-0-82
Degree $2$
Conductor $1183$
Sign $-0.761 + 0.647i$
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  + (0.603 − 1.04i)2-s + (−0.516 − 0.893i)3-s + (0.272 + 0.472i)4-s + (1.51 − 2.62i)5-s − 1.24·6-s + (1.04 − 2.43i)7-s + 3.06·8-s + (0.967 − 1.67i)9-s + (−1.82 − 3.16i)10-s + (−2.01 − 3.48i)11-s + (0.281 − 0.487i)12-s + (−1.90 − 2.55i)14-s − 3.12·15-s + (1.30 − 2.26i)16-s + (2.78 + 4.82i)17-s + (−1.16 − 2.02i)18-s + ⋯
L(s)  = 1  + (0.426 − 0.738i)2-s + (−0.297 − 0.516i)3-s + (0.136 + 0.236i)4-s + (0.677 − 1.17i)5-s − 0.508·6-s + (0.394 − 0.918i)7-s + 1.08·8-s + (0.322 − 0.558i)9-s + (−0.577 − 1.00i)10-s + (−0.606 − 1.05i)11-s + (0.0812 − 0.140i)12-s + (−0.510 − 0.683i)14-s − 0.807·15-s + (0.326 − 0.565i)16-s + (0.675 + 1.16i)17-s + (−0.275 − 0.476i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.761 + 0.647i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.761 + 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.761 + 0.647i$
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: \((2,\ 1183,\ (\ :1/2),\ -0.761 + 0.647i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.506571470\)
\(L(\frac12)\) \(\approx\) \(2.506571470\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-1.04 + 2.43i)T \)
13 \( 1 \)
good2 \( 1 + (-0.603 + 1.04i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (0.516 + 0.893i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.51 + 2.62i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.01 + 3.48i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.78 - 4.82i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.186 + 0.322i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.43 - 7.67i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.34T + 29T^{2} \)
31 \( 1 + (-4.03 - 6.98i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.80 + 3.12i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 5.05T + 41T^{2} \)
43 \( 1 - 0.00422T + 43T^{2} \)
47 \( 1 + (2.07 - 3.58i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.20 + 7.28i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.42 - 2.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.55 - 2.69i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.71 + 2.96i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.19T + 71T^{2} \)
73 \( 1 + (-7.04 - 12.1i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.31 - 7.47i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.23T + 83T^{2} \)
89 \( 1 + (0.850 - 1.47i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 2.01T + 97T^{2} \)
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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

−9.732916602139837657520731610505, −8.405115261624709764434806019924, −7.980204954479533983119017654669, −6.94589100314908258034673179743, −5.91756601922939834682390615159, −5.09527723614688607912734158221, −4.07883925115554479448313297176, −3.26027767326435719400622043121, −1.64116606828670144603051899246, −1.08033113927660118237367047955, 2.03127523121340652140129467459, 2.69688595484839201321153283767, 4.56970547428129685246498792770, 4.97587159384302338196092880714, 5.94368531997566123784353881145, 6.53326331233516543036722910905, 7.45925987554102165666587490008, 8.125458221147760688805798825823, 9.639381692602120887914781796022, 10.16154050219716883392230282078

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