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Properties

Label 2-1183-7.2-c1-0-86
Degree $2$
Conductor $1183$
Sign $-0.963 - 0.267i$
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  + (1.18 − 2.05i)2-s + (0.348 + 0.603i)3-s + (−1.82 − 3.15i)4-s + (1.31 − 2.27i)5-s + 1.65·6-s + (−2.59 + 0.497i)7-s − 3.90·8-s + (1.25 − 2.17i)9-s + (−3.11 − 5.40i)10-s + (−2.23 − 3.86i)11-s + (1.26 − 2.19i)12-s + (−2.06 + 5.93i)14-s + 1.82·15-s + (−0.991 + 1.71i)16-s + (1.17 + 2.03i)17-s + (−2.98 − 5.17i)18-s + ⋯
L(s)  = 1  + (0.839 − 1.45i)2-s + (0.201 + 0.348i)3-s + (−0.910 − 1.57i)4-s + (0.587 − 1.01i)5-s + 0.675·6-s + (−0.982 + 0.188i)7-s − 1.37·8-s + (0.419 − 0.725i)9-s + (−0.986 − 1.70i)10-s + (−0.673 − 1.16i)11-s + (0.366 − 0.634i)12-s + (−0.551 + 1.58i)14-s + 0.472·15-s + (−0.247 + 0.429i)16-s + (0.285 + 0.494i)17-s + (−0.704 − 1.21i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.963 - 0.267i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.963 - 0.267i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.963 - 0.267i$
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: \((2,\ 1183,\ (\ :1/2),\ -0.963 - 0.267i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.306268081\)
\(L(\frac12)\) \(\approx\) \(2.306268081\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.59 - 0.497i)T \)
13 \( 1 \)
good2 \( 1 + (-1.18 + 2.05i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-0.348 - 0.603i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.31 + 2.27i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.23 + 3.86i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.17 - 2.03i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.41 - 5.90i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.955 - 1.65i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.24T + 29T^{2} \)
31 \( 1 + (3.77 + 6.54i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.97 + 5.15i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 0.271T + 41T^{2} \)
43 \( 1 + 3.28T + 43T^{2} \)
47 \( 1 + (4.05 - 7.02i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.59 + 6.22i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.52 + 7.83i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.94 - 6.83i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.50 - 6.06i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.84T + 71T^{2} \)
73 \( 1 + (-0.174 - 0.301i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.26 + 10.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 13.4T + 83T^{2} \)
89 \( 1 + (-3.00 + 5.20i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 12.6T + 97T^{2} \)
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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

−9.624965304785701666543835715646, −8.939379690084361502739779182089, −7.999185572435865592537071972971, −6.19607804750139382697945826095, −5.78336757745040000768148565637, −4.73831613595308292217944768382, −3.77169973763008918530090613740, −3.20117342683276204168803128083, −1.96089495673634194848350942505, −0.72046008055617917598197385375, 2.31797678347771429616107075553, 3.16545996110493327331665524867, 4.57769998461275112546459282166, 5.11384553635345006517224256381, 6.52697383940509737474541690303, 6.64126848124036265856429649100, 7.35891923144475707939357692734, 8.128953569135017786106092770549, 9.257922938424063616331549698764, 10.28159498909041145290895343875

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