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Properties

Label 2-1183-7.2-c1-0-64
Degree $2$
Conductor $1183$
Sign $0.999 - 0.0144i$
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  + (−1.06 + 1.84i)2-s + (0.0894 + 0.154i)3-s + (−1.25 − 2.18i)4-s + (1.80 − 3.12i)5-s − 0.380·6-s + (2.35 − 1.20i)7-s + 1.10·8-s + (1.48 − 2.57i)9-s + (3.83 + 6.63i)10-s + (1.99 + 3.45i)11-s + (0.225 − 0.389i)12-s + (−0.274 + 5.61i)14-s + 0.644·15-s + (1.34 − 2.33i)16-s + (−2.39 − 4.14i)17-s + (3.15 + 5.46i)18-s + ⋯
L(s)  = 1  + (−0.751 + 1.30i)2-s + (0.0516 + 0.0894i)3-s + (−0.629 − 1.09i)4-s + (0.806 − 1.39i)5-s − 0.155·6-s + (0.889 − 0.457i)7-s + 0.389·8-s + (0.494 − 0.856i)9-s + (1.21 + 2.09i)10-s + (0.601 + 1.04i)11-s + (0.0649 − 0.112i)12-s + (−0.0734 + 1.50i)14-s + 0.166·15-s + (0.337 − 0.583i)16-s + (−0.580 − 1.00i)17-s + (0.743 + 1.28i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0144i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0144i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.999 - 0.0144i$
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: \((2,\ 1183,\ (\ :1/2),\ 0.999 - 0.0144i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.366932852\)
\(L(\frac12)\) \(\approx\) \(1.366932852\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.35 + 1.20i)T \)
13 \( 1 \)
good2 \( 1 + (1.06 - 1.84i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-0.0894 - 0.154i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.80 + 3.12i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.99 - 3.45i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.39 + 4.14i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.57 - 2.72i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.08 + 1.88i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.57T + 29T^{2} \)
31 \( 1 + (-0.743 - 1.28i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.48 + 4.29i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.11T + 41T^{2} \)
43 \( 1 - 1.43T + 43T^{2} \)
47 \( 1 + (-0.509 + 0.882i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.01 + 5.22i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.45 + 4.24i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.01 + 1.76i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.95 - 3.38i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.80T + 71T^{2} \)
73 \( 1 + (-1.54 - 2.67i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.984 - 1.70i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 7.66T + 83T^{2} \)
89 \( 1 + (6.39 - 11.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 1.35T + 97T^{2} \)
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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

−9.489859213732309744563111525364, −8.944387092030371944847101205399, −8.193669906440909340215105520587, −7.28236853130385018849290956474, −6.64892317821221437115001653393, −5.63616171232063055159877732574, −4.85073354074536392159348591437, −4.10679106886881011981580975115, −1.88297486101662372058179760125, −0.805419249869439120231882774708, 1.53007624959230023733890985424, 2.23012306559147992797707403618, 3.06575718157812501524274512556, 4.21459302840514893297561530080, 5.67577927505266238147508283255, 6.40351958786700040393896445886, 7.52216353942317239797804533108, 8.415677456762584327837050174795, 9.137840849267114562936422785270, 9.946095274236452019874167995566

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