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Properties

Label 2-1183-7.2-c1-0-81
Degree $2$
Conductor $1183$
Sign $-0.171 + 0.985i$
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  + (−0.425 + 0.737i)2-s + (−0.330 − 0.572i)3-s + (0.637 + 1.10i)4-s + (1.72 − 2.98i)5-s + 0.562·6-s + (−0.751 − 2.53i)7-s − 2.78·8-s + (1.28 − 2.21i)9-s + (1.46 + 2.53i)10-s + (−0.448 − 0.777i)11-s + (0.421 − 0.730i)12-s + (2.18 + 0.525i)14-s − 2.27·15-s + (−0.0891 + 0.154i)16-s + (−0.968 − 1.67i)17-s + (1.09 + 1.88i)18-s + ⋯
L(s)  = 1  + (−0.300 + 0.521i)2-s + (−0.190 − 0.330i)3-s + (0.318 + 0.552i)4-s + (0.769 − 1.33i)5-s + 0.229·6-s + (−0.284 − 0.958i)7-s − 0.985·8-s + (0.427 − 0.739i)9-s + (0.463 + 0.802i)10-s + (−0.135 − 0.234i)11-s + (0.121 − 0.210i)12-s + (0.585 + 0.140i)14-s − 0.587·15-s + (−0.0222 + 0.0386i)16-s + (−0.234 − 0.406i)17-s + (0.257 + 0.445i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.171 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.171 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.171 + 0.985i$
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: \((2,\ 1183,\ (\ :1/2),\ -0.171 + 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.168185318\)
\(L(\frac12)\) \(\approx\) \(1.168185318\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.751 + 2.53i)T \)
13 \( 1 \)
good2 \( 1 + (0.425 - 0.737i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (0.330 + 0.572i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.72 + 2.98i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.448 + 0.777i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.968 + 1.67i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.519 + 0.898i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.82 - 4.89i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 1.83T + 29T^{2} \)
31 \( 1 + (4.56 + 7.91i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.30 - 9.17i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 5.33T + 41T^{2} \)
43 \( 1 + 3.91T + 43T^{2} \)
47 \( 1 + (-3.59 + 6.22i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.69 - 8.12i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.255 + 0.442i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.718 - 1.24i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.22 + 7.31i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.44T + 71T^{2} \)
73 \( 1 + (-5.45 - 9.44i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.04 + 10.4i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.51T + 83T^{2} \)
89 \( 1 + (-6.80 + 11.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 0.506T + 97T^{2} \)
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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

−9.401640595156719338884609320853, −8.757230509833018493078045922426, −7.75577638907886030690529680768, −7.15716370354588775662626678419, −6.25915381532148339571176748883, −5.55726816804658317076161255192, −4.34107595152188019954742836366, −3.39994522758848791741213359748, −1.80335420809585413847257725774, −0.52973966593910701477704331081, 1.97140952877369954648852595974, 2.38782666609851711152030244412, 3.54007927022799932610087802109, 5.11477876603124269436012616633, 5.84631269199609815280068900524, 6.52849609656611175121189507301, 7.34446488027811485440022632823, 8.666487765960607851054234557671, 9.507563159065499340622900171839, 10.15621778403875290672950647819

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