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Properties

Label 1-1183-1183.144-r0-0-0
Degree $1$
Conductor $1183$
Sign $0.350 - 0.936i$
Analytic cond. $5.49382$
Root an. cond. $5.49382$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.948 − 0.316i)2-s + (0.987 − 0.160i)3-s + (0.799 − 0.600i)4-s + (0.692 − 0.721i)5-s + (0.885 − 0.464i)6-s + (0.568 − 0.822i)8-s + (0.948 − 0.316i)9-s + (0.428 − 0.903i)10-s + (0.948 + 0.316i)11-s + (0.692 − 0.721i)12-s + (0.568 − 0.822i)15-s + (0.278 − 0.960i)16-s + (−0.996 − 0.0804i)17-s + (0.799 − 0.600i)18-s + (−0.5 + 0.866i)19-s + (0.120 − 0.992i)20-s + ⋯
L(s)  = 1  + (0.948 − 0.316i)2-s + (0.987 − 0.160i)3-s + (0.799 − 0.600i)4-s + (0.692 − 0.721i)5-s + (0.885 − 0.464i)6-s + (0.568 − 0.822i)8-s + (0.948 − 0.316i)9-s + (0.428 − 0.903i)10-s + (0.948 + 0.316i)11-s + (0.692 − 0.721i)12-s + (0.568 − 0.822i)15-s + (0.278 − 0.960i)16-s + (−0.996 − 0.0804i)17-s + (0.799 − 0.600i)18-s + (−0.5 + 0.866i)19-s + (0.120 − 0.992i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.350 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.350 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.350 - 0.936i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (144, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ 0.350 - 0.936i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.907023955 - 2.709849501i\)
\(L(\frac12)\) \(\approx\) \(3.907023955 - 2.709849501i\)
\(L(1)\) \(\approx\) \(2.646565833 - 1.112390651i\)
\(L(1)\) \(\approx\) \(2.646565833 - 1.112390651i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.948 - 0.316i)T \)
3 \( 1 + (0.987 - 0.160i)T \)
5 \( 1 + (0.692 - 0.721i)T \)
11 \( 1 + (0.948 + 0.316i)T \)
17 \( 1 + (-0.996 - 0.0804i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.748 - 0.663i)T \)
31 \( 1 + (-0.845 - 0.534i)T \)
37 \( 1 + (-0.0402 + 0.999i)T \)
41 \( 1 + (-0.354 + 0.935i)T \)
43 \( 1 + (0.885 + 0.464i)T \)
47 \( 1 + (0.799 + 0.600i)T \)
53 \( 1 + (-0.996 - 0.0804i)T \)
59 \( 1 + (0.278 + 0.960i)T \)
61 \( 1 + (-0.996 + 0.0804i)T \)
67 \( 1 + (-0.919 + 0.391i)T \)
71 \( 1 + (-0.354 + 0.935i)T \)
73 \( 1 + (0.948 + 0.316i)T \)
79 \( 1 + (0.799 + 0.600i)T \)
83 \( 1 + (-0.354 - 0.935i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.970 + 0.239i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.57129048597559780718139021713, −20.717789073822609863712659056505, −19.97033751087254025627629828319, −19.33411658464232992665006256405, −18.30399317463483842078156621910, −17.4214316357545048947983714777, −16.5729470404660619250598109486, −15.6365932939934062733951876217, −14.9753560054500368545912580686, −14.27798812878573795510860977597, −13.859758995938635571842840052190, −13.05124222805639234745430812431, −12.29591940376188401328230232696, −10.9559243227131855353188221558, −10.647831366029787908254198151513, −9.207194300941867632700118822201, −8.78973747976357829666782797806, −7.51436003641440662290063157324, −6.83104365839434192517258071992, −6.18701607201630819434941499800, −5.043557242929939271886029733652, −4.04291553594968820980940905059, −3.398681542451116018394586312592, −2.36570891655317966275730586475, −1.83967008793686282748811782550, 1.40051005111205094599538564813, 1.8674247535119869054299372341, 2.851187093471718401338920958972, 4.063729733946015843950665670995, 4.38951466358587541120100164747, 5.705882069783256885650657126875, 6.40089306892988587127577169310, 7.36776610477527450771038637334, 8.36676951106941833478335003578, 9.45325734378386235998366865949, 9.73839994090291919390024249835, 10.95262884850427416701860509284, 11.98602626521013135563970465839, 12.67313113865946288580405429347, 13.38655008071903728692673112349, 13.90319900261054548571593916738, 14.723532691333377930293486081489, 15.33233738081058506524314946843, 16.28626242735640878338967031995, 17.10997054251514256551352608607, 18.12946773264872063592502960064, 19.13155352531646399677945231674, 19.8020600320705844056285771437, 20.4571674597646567121698245417, 20.90339242650943440933993796137

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