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Properties

Label 2-1183-7.2-c1-0-20
Degree $2$
Conductor $1183$
Sign $-0.605 + 0.795i$
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  + (−0.5 + 0.866i)2-s + (1.5 + 2.59i)3-s + (0.500 + 0.866i)4-s + (−1.5 + 2.59i)5-s − 3·6-s + (0.5 − 2.59i)7-s − 3·8-s + (−3 + 5.19i)9-s + (−1.5 − 2.59i)10-s + (1.5 + 2.59i)11-s + (−1.49 + 2.59i)12-s + (2 + 1.73i)14-s − 9·15-s + (0.500 − 0.866i)16-s + (1 + 1.73i)17-s + (−3 − 5.19i)18-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.866 + 1.49i)3-s + (0.250 + 0.433i)4-s + (−0.670 + 1.16i)5-s − 1.22·6-s + (0.188 − 0.981i)7-s − 1.06·8-s + (−1 + 1.73i)9-s + (−0.474 − 0.821i)10-s + (0.452 + 0.783i)11-s + (−0.433 + 0.749i)12-s + (0.534 + 0.462i)14-s − 2.32·15-s + (0.125 − 0.216i)16-s + (0.242 + 0.420i)17-s + (−0.707 − 1.22i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.605 + 0.795i$
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: \((2,\ 1183,\ (\ :1/2),\ -0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.538855597\)
\(L(\frac12)\) \(\approx\) \(1.538855597\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.5 + 2.59i)T \)
13 \( 1 \)
good2 \( 1 + (0.5 - 0.866i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-1.5 - 2.59i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.5 - 2.59i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 7T + 29T^{2} \)
31 \( 1 + (1.5 + 2.59i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 3T + 41T^{2} \)
43 \( 1 + 7T + 43T^{2} \)
47 \( 1 + (0.5 - 0.866i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.5 + 2.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.5 + 11.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.5 - 2.59i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 13T + 71T^{2} \)
73 \( 1 + (-6.5 - 11.2i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.5 + 2.59i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 5T + 97T^{2} \)
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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

−10.06079988770602998195650327371, −9.589472085088987738886881322831, −8.457303233765370896824918830598, −7.960343463594599175013528876389, −7.13384198679466693510945879505, −6.51996515340616316203865059603, −4.92776849126758214528802585870, −3.90684296154350397270770949835, −3.50523013181163719205874532464, −2.51767491301095379565245519379, 0.68640472950616425433565807329, 1.50910939711207774995178201881, 2.53984078477029957018061134867, 3.44397887691165123366912573880, 5.06524301912545564168067974061, 6.01276094989407434318296635795, 6.81035681456118698446561915609, 7.924630696192998327259565704011, 8.601413937958611170742838679339, 8.880551276128510091210319388905

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