Properties

Label 2-1183-7.2-c1-0-63
Degree $2$
Conductor $1183$
Sign $0.0836 + 0.996i$
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  + (0.166 − 0.287i)2-s + (−0.729 − 1.26i)3-s + (0.944 + 1.63i)4-s + (0.722 − 1.25i)5-s − 0.485·6-s + (1.36 + 2.26i)7-s + 1.29·8-s + (0.434 − 0.752i)9-s + (−0.240 − 0.416i)10-s + (−2.97 − 5.15i)11-s + (1.37 − 2.38i)12-s + (0.879 − 0.0178i)14-s − 2.11·15-s + (−1.67 + 2.90i)16-s + (−2.16 − 3.74i)17-s + (−0.144 − 0.250i)18-s + ⋯
L(s)  = 1  + (0.117 − 0.203i)2-s + (−0.421 − 0.729i)3-s + (0.472 + 0.818i)4-s + (0.323 − 0.559i)5-s − 0.198·6-s + (0.517 + 0.855i)7-s + 0.457·8-s + (0.144 − 0.250i)9-s + (−0.0759 − 0.131i)10-s + (−0.897 − 1.55i)11-s + (0.398 − 0.689i)12-s + (0.234 − 0.00477i)14-s − 0.544·15-s + (−0.418 + 0.725i)16-s + (−0.524 − 0.909i)17-s + (−0.0340 − 0.0589i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0836 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0836 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.0836 + 0.996i$
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: \((2,\ 1183,\ (\ :1/2),\ 0.0836 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.776785610\)
\(L(\frac12)\) \(\approx\) \(1.776785610\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-1.36 - 2.26i)T \)
13 \( 1 \)
good2 \( 1 + (-0.166 + 0.287i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (0.729 + 1.26i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.722 + 1.25i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.97 + 5.15i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.16 + 3.74i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.978 + 1.69i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.270 + 0.467i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.15T + 29T^{2} \)
31 \( 1 + (3.05 + 5.28i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.01 + 6.95i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 7.55T + 41T^{2} \)
43 \( 1 - 4.24T + 43T^{2} \)
47 \( 1 + (-3.13 + 5.42i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.38 - 2.40i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.425 + 0.737i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.38 - 5.87i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.493 + 0.854i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 3.76T + 71T^{2} \)
73 \( 1 + (4.56 + 7.91i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.0655 + 0.113i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 2.66T + 83T^{2} \)
89 \( 1 + (-4.85 + 8.41i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 6.58T + 97T^{2} \)
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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

−9.270556438210190386321220511815, −8.745896161002122746847788663876, −7.85661736955564621744947961314, −7.19933058574956713650125181625, −6.08822757890254937830481692557, −5.52708556273154297926393406737, −4.41142140902429097860704755139, −3.02488193450650729381999441810, −2.24684739482886339116995408781, −0.798037528409199423036396436205, 1.52694886330486352667732215955, 2.56933138737901136143411022204, 4.33749257192171592176992444954, 4.72564422100994104377879438585, 5.65887159574477992535039066015, 6.64233123416908606940913027110, 7.31454862785523232913017037892, 8.102996401842178865310406331828, 9.646284981577355266946488361670, 10.14616871166661509751192741495

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