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Properties

Label 2-1183-7.4-c1-0-41
Degree $2$
Conductor $1183$
Sign $-0.991 - 0.130i$
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  + (1.36 + 2.36i)2-s + (0.673 − 1.16i)3-s + (−2.71 + 4.69i)4-s + (1.09 + 1.89i)5-s + 3.66·6-s + (2.19 + 1.47i)7-s − 9.33·8-s + (0.593 + 1.02i)9-s + (−2.98 + 5.16i)10-s + (−0.524 + 0.907i)11-s + (3.65 + 6.32i)12-s + (−0.484 + 7.19i)14-s + 2.94·15-s + (−7.29 − 12.6i)16-s + (2.64 − 4.58i)17-s + (−1.61 + 2.80i)18-s + ⋯
L(s)  = 1  + (0.963 + 1.66i)2-s + (0.388 − 0.673i)3-s + (−1.35 + 2.34i)4-s + (0.489 + 0.847i)5-s + 1.49·6-s + (0.830 + 0.557i)7-s − 3.30·8-s + (0.197 + 0.342i)9-s + (−0.942 + 1.63i)10-s + (−0.158 + 0.273i)11-s + (1.05 + 1.82i)12-s + (−0.129 + 1.92i)14-s + 0.760·15-s + (−1.82 − 3.16i)16-s + (0.641 − 1.11i)17-s + (−0.381 + 0.660i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.130i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.991 - 0.130i$
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: \((2,\ 1183,\ (\ :1/2),\ -0.991 - 0.130i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.231447632\)
\(L(\frac12)\) \(\approx\) \(3.231447632\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.19 - 1.47i)T \)
13 \( 1 \)
good2 \( 1 + (-1.36 - 2.36i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.673 + 1.16i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.09 - 1.89i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.524 - 0.907i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.64 + 4.58i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.378 - 0.655i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.326 + 0.566i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.10T + 29T^{2} \)
31 \( 1 + (-0.513 + 0.890i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.44 + 9.43i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 7.32T + 41T^{2} \)
43 \( 1 - 0.887T + 43T^{2} \)
47 \( 1 + (-1.16 - 2.02i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.44 - 4.23i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.524 - 0.907i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.24 - 10.8i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.23 + 3.87i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.60T + 71T^{2} \)
73 \( 1 + (4.14 - 7.17i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.07 + 1.85i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6.66T + 83T^{2} \)
89 \( 1 + (2.88 + 4.99i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 2.88T + 97T^{2} \)
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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

−9.904134907134233172180196613405, −8.838076874984883636790511634460, −8.153970935715507794690770196023, −7.26079201018480964194537884935, −7.12247315923743586927921456165, −5.94449818976102534770993433707, −5.31630044189502091834854137579, −4.46436200292134322275990222020, −3.14082863159984650513535347444, −2.20831483594276703939584522927, 1.07337059955423013253447857567, 1.87341689559828041142864780259, 3.35349231667439640073397596676, 3.88894307702409802039179249506, 4.91122921258198563998998131638, 5.26194533156077861346746754283, 6.47359161461101039552038908704, 8.220322532797801705357406876210, 8.922118673917951903893176814867, 9.750516152869429895139976960843

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