Properties

Label 2-1183-7.4-c1-0-18
Degree $2$
Conductor $1183$
Sign $0.328 + 0.944i$
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  + (1.14 + 1.97i)2-s + (−1.57 + 2.72i)3-s + (−1.61 + 2.78i)4-s + (1.06 + 1.84i)5-s − 7.19·6-s + (−0.331 + 2.62i)7-s − 2.78·8-s + (−3.46 − 5.99i)9-s + (−2.42 + 4.20i)10-s + (0.154 − 0.267i)11-s + (−5.07 − 8.78i)12-s + (−5.57 + 2.34i)14-s − 6.69·15-s + (0.0349 + 0.0605i)16-s + (0.887 − 1.53i)17-s + (7.91 − 13.7i)18-s + ⋯
L(s)  = 1  + (0.807 + 1.39i)2-s + (−0.909 + 1.57i)3-s + (−0.805 + 1.39i)4-s + (0.475 + 0.823i)5-s − 2.93·6-s + (−0.125 + 0.992i)7-s − 0.985·8-s + (−1.15 − 1.99i)9-s + (−0.767 + 1.32i)10-s + (0.0465 − 0.0805i)11-s + (−1.46 − 2.53i)12-s + (−1.48 + 0.626i)14-s − 1.72·15-s + (0.00874 + 0.0151i)16-s + (0.215 − 0.372i)17-s + (1.86 − 3.22i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.328 + 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.328 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.328 + 0.944i$
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: \((2,\ 1183,\ (\ :1/2),\ 0.328 + 0.944i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.699749837\)
\(L(\frac12)\) \(\approx\) \(1.699749837\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.331 - 2.62i)T \)
13 \( 1 \)
good2 \( 1 + (-1.14 - 1.97i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (1.57 - 2.72i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.06 - 1.84i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.154 + 0.267i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.887 + 1.53i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.890 - 1.54i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.575 + 0.996i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.01T + 29T^{2} \)
31 \( 1 + (2.30 - 3.98i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.77 + 4.79i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 6.72T + 41T^{2} \)
43 \( 1 - 1.52T + 43T^{2} \)
47 \( 1 + (-4.75 - 8.24i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.72 - 6.44i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.06 - 7.03i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.72 - 2.97i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.30 + 10.9i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.35T + 71T^{2} \)
73 \( 1 + (-5.94 + 10.2i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.96 - 6.86i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 11.2T + 83T^{2} \)
89 \( 1 + (0.829 + 1.43i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 7.66T + 97T^{2} \)
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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

−10.50965376940978128988100092716, −9.474841833580518697655021590922, −8.934773736242409912390341707434, −7.74299133834160168377974384317, −6.57550474121535743528824877755, −6.06333244905865349380037449889, −5.45945980612148887679606102580, −4.77451069946415454920354013930, −3.82211574267717248385639632801, −2.85266956469908235335592050990, 0.67609476700361795691577898453, 1.41070396991921967163025636256, 2.32245602013049890425339628067, 3.71919323819010724044371807109, 4.87203878156146389207158108455, 5.46638987669940897354284536602, 6.42562728544785030884785840813, 7.30878927686153365662132850031, 8.159686076129320065983040663789, 9.433161084724937379935085514081

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