Properties

Label 2-1183-7.4-c1-0-61
Degree $2$
Conductor $1183$
Sign $0.936 - 0.349i$
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  + (1.10 + 1.91i)2-s + (−1.23 + 2.14i)3-s + (−1.44 + 2.51i)4-s + (−1.06 − 1.83i)5-s − 5.47·6-s + (−2.63 − 0.272i)7-s − 1.98·8-s + (−1.56 − 2.70i)9-s + (2.34 − 4.06i)10-s + (2.39 − 4.14i)11-s + (−3.58 − 6.21i)12-s + (−2.39 − 5.34i)14-s + 5.25·15-s + (0.697 + 1.20i)16-s + (1.88 − 3.27i)17-s + (3.45 − 5.98i)18-s + ⋯
L(s)  = 1  + (0.782 + 1.35i)2-s + (−0.714 + 1.23i)3-s + (−0.724 + 1.25i)4-s + (−0.474 − 0.822i)5-s − 2.23·6-s + (−0.994 − 0.102i)7-s − 0.703·8-s + (−0.520 − 0.901i)9-s + (0.742 − 1.28i)10-s + (0.721 − 1.25i)11-s + (−1.03 − 1.79i)12-s + (−0.638 − 1.42i)14-s + 1.35·15-s + (0.174 + 0.301i)16-s + (0.458 − 0.793i)17-s + (0.814 − 1.41i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 - 0.349i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.936 - 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.936 - 0.349i$
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: \((2,\ 1183,\ (\ :1/2),\ 0.936 - 0.349i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9113119769\)
\(L(\frac12)\) \(\approx\) \(0.9113119769\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.63 + 0.272i)T \)
13 \( 1 \)
good2 \( 1 + (-1.10 - 1.91i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (1.23 - 2.14i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.06 + 1.83i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.39 + 4.14i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.88 + 3.27i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.78 + 3.08i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.23 + 3.87i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 5.90T + 29T^{2} \)
31 \( 1 + (1.88 - 3.26i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.81 - 4.87i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 10.3T + 41T^{2} \)
43 \( 1 - 3.40T + 43T^{2} \)
47 \( 1 + (3.55 + 6.15i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.19 + 10.7i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.39 + 4.14i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.60 + 2.77i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.44 - 2.51i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.53T + 71T^{2} \)
73 \( 1 + (-3.85 + 6.66i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.58 - 4.48i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.46T + 83T^{2} \)
89 \( 1 + (-1.83 - 3.17i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 5.40T + 97T^{2} \)
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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

−9.675730217133915048019770454897, −8.844289181046552414267888727175, −8.217619244184841499248278091064, −6.93265087279004216808974993794, −6.32027578696684695382040375672, −5.44350548880500205281357901110, −4.88515280303707583172008215344, −3.98962305337986467696424071924, −3.41244324716963943932487955176, −0.34413024237294033164936519996, 1.42683207716793274764940884928, 2.26397379009176753480227391705, 3.51974463214059652768854227097, 4.08494525304162319241022525522, 5.58573959653048706305825860576, 6.24791979831468884559565892798, 7.19081076619295672696319919050, 7.64684001595271913907805246398, 9.316805510009513251602714750401, 10.05833357760447310923690573268

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