Properties

Label 2-1183-7.4-c1-0-65
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

\(L(1)\) \(\approx\) \(0.06793491335\)
\(L(\frac12)\) \(\approx\) \(0.06793491335\)
\(L(\frac{3}{2})\) 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

−9.649005064070594683803375946873, −8.733683669356037186783054651434, −8.104331772156393075801597817338, −6.66879954307783229429939509190, −5.35370010222258194001810071076, −4.42243952148097991666528858853, −4.03407545088549865090262221867, −2.95708936891548779282119924635, −1.03550794899203542288566806224, −0.05527137408281841379879947566, 1.55234914789366846970366344476, 2.94438594087667408146351063724, 5.02098289040479679968184451686, 5.84079707933717060461286690266, 6.38900912608808432153272359748, 6.98902378082826442032396025485, 7.79312748843890748432988911234, 8.197377999078605488732586733909, 9.117993889727545725626307043568, 10.32040246637544998836019064014

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