Properties

Label 2-1183-7.4-c1-0-20
Degree $2$
Conductor $1183$
Sign $-0.458 + 0.888i$
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  + (1.29 + 2.24i)2-s + (−0.259 + 0.449i)3-s + (−2.35 + 4.07i)4-s + (0.806 + 1.39i)5-s − 1.34·6-s + (−2.13 + 1.56i)7-s − 6.99·8-s + (1.36 + 2.36i)9-s + (−2.08 + 3.61i)10-s + (1.35 − 2.34i)11-s + (−1.21 − 2.11i)12-s + (−6.27 − 2.74i)14-s − 0.835·15-s + (−4.34 − 7.53i)16-s + (−1.56 + 2.70i)17-s + (−3.53 + 6.12i)18-s + ⋯
L(s)  = 1  + (0.915 + 1.58i)2-s + (−0.149 + 0.259i)3-s + (−1.17 + 2.03i)4-s + (0.360 + 0.624i)5-s − 0.547·6-s + (−0.805 + 0.592i)7-s − 2.47·8-s + (0.455 + 0.788i)9-s + (−0.659 + 1.14i)10-s + (0.407 − 0.706i)11-s + (−0.351 − 0.609i)12-s + (−1.67 − 0.734i)14-s − 0.215·15-s + (−1.08 − 1.88i)16-s + (−0.379 + 0.656i)17-s + (−0.833 + 1.44i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.458 + 0.888i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.458 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.458 + 0.888i$
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: \((2,\ 1183,\ (\ :1/2),\ -0.458 + 0.888i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.063019531\)
\(L(\frac12)\) \(\approx\) \(2.063019531\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.13 - 1.56i)T \)
13 \( 1 \)
good2 \( 1 + (-1.29 - 2.24i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.259 - 0.449i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.806 - 1.39i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.35 + 2.34i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.56 - 2.70i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.84 - 3.18i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.993 + 1.71i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 5.37T + 29T^{2} \)
31 \( 1 + (-5.23 + 9.07i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.97 - 5.15i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 7.70T + 41T^{2} \)
43 \( 1 + 3.35T + 43T^{2} \)
47 \( 1 + (-0.527 - 0.913i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.63 - 6.29i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.71 + 9.89i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.46 - 2.53i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.79 - 11.7i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.35T + 71T^{2} \)
73 \( 1 + (-4.55 + 7.88i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.10 - 5.37i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.69T + 83T^{2} \)
89 \( 1 + (-0.879 - 1.52i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 15.4T + 97T^{2} \)
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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

−10.11735162417603551808062081729, −9.347312425633898528488821283384, −8.325741658498016423659631995002, −7.70697232289672876680702134880, −6.65410322607046301228489395066, −6.11592191124507441424190215581, −5.58590237280395326341211049941, −4.43994862474449808252232384355, −3.65923091541204209077222528671, −2.52384537505261465767407251119, 0.69105622289586672664483522891, 1.63420438444421545188188892238, 2.92362086219957227438350503518, 3.83961696628356078387323010502, 4.60991557344420220507448212310, 5.46947786930391089726825034409, 6.53043791091377700196024390677, 7.28645616699679557835748285240, 9.118798291348273802590942659797, 9.420959083072459513975170264359

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