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Properties

Label 1-1183-1183.489-r0-0-0
Degree $1$
Conductor $1183$
Sign $0.703 - 0.710i$
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  + (0.822 + 0.568i)2-s + (−0.885 + 0.464i)3-s + (0.354 + 0.935i)4-s + (−0.663 + 0.748i)5-s + (−0.992 − 0.120i)6-s + (−0.239 + 0.970i)8-s + (0.568 − 0.822i)9-s + (−0.970 + 0.239i)10-s + (0.822 − 0.568i)11-s + (−0.748 − 0.663i)12-s + (0.239 − 0.970i)15-s + (−0.748 + 0.663i)16-s + (−0.970 − 0.239i)17-s + (0.935 − 0.354i)18-s − i·19-s + (−0.935 − 0.354i)20-s + ⋯
L(s)  = 1  + (0.822 + 0.568i)2-s + (−0.885 + 0.464i)3-s + (0.354 + 0.935i)4-s + (−0.663 + 0.748i)5-s + (−0.992 − 0.120i)6-s + (−0.239 + 0.970i)8-s + (0.568 − 0.822i)9-s + (−0.970 + 0.239i)10-s + (0.822 − 0.568i)11-s + (−0.748 − 0.663i)12-s + (0.239 − 0.970i)15-s + (−0.748 + 0.663i)16-s + (−0.970 − 0.239i)17-s + (0.935 − 0.354i)18-s − i·19-s + (−0.935 − 0.354i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.703 - 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.703 - 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.703 - 0.710i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (489, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ 0.703 - 0.710i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4636434579 - 0.1933108221i\)
\(L(\frac12)\) \(\approx\) \(0.4636434579 - 0.1933108221i\)
\(L(1)\) \(\approx\) \(0.8055436070 + 0.4282113544i\)
\(L(1)\) \(\approx\) \(0.8055436070 + 0.4282113544i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.822 + 0.568i)T \)
3 \( 1 + (-0.885 + 0.464i)T \)
5 \( 1 + (-0.663 + 0.748i)T \)
11 \( 1 + (0.822 - 0.568i)T \)
17 \( 1 + (-0.970 - 0.239i)T \)
19 \( 1 - iT \)
23 \( 1 - T \)
29 \( 1 + (0.568 - 0.822i)T \)
31 \( 1 + (-0.992 - 0.120i)T \)
37 \( 1 + (-0.992 - 0.120i)T \)
41 \( 1 + (-0.464 - 0.885i)T \)
43 \( 1 + (-0.120 - 0.992i)T \)
47 \( 1 + (-0.935 - 0.354i)T \)
53 \( 1 + (-0.970 - 0.239i)T \)
59 \( 1 + (0.663 - 0.748i)T \)
61 \( 1 + (0.970 - 0.239i)T \)
67 \( 1 + (-0.935 - 0.354i)T \)
71 \( 1 + (0.464 + 0.885i)T \)
73 \( 1 + (-0.822 + 0.568i)T \)
79 \( 1 + (-0.354 + 0.935i)T \)
83 \( 1 + (0.464 - 0.885i)T \)
89 \( 1 + iT \)
97 \( 1 + (0.663 + 0.748i)T \)
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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

−21.44004624357597625056364743666, −20.496945899394625845181410111824, −19.79076121902500133625541226028, −19.32067408420632366064305763041, −18.2776105747533821651891082339, −17.563804553112259681598220861313, −16.44029015687355215488578655890, −16.0693760634881770016615664142, −15.03567231592119910383557813751, −14.22445718444636468906867095160, −13.18058776617782223506118427326, −12.615316830689935224301252589, −11.97126671259003968518480079279, −11.46915931210731304206175924096, −10.56354562497567260954550225891, −9.70774328261523770230183657395, −8.60831891446402336425509245928, −7.494139610296583861254808154668, −6.62088035081985020350916830022, −5.89375923397291814520358612200, −4.86766607803552863100563428590, −4.35463815753643474985455490359, −3.44149187354278993631056967731, −1.843091058483529180484186078723, −1.3533094426050986595087596610, 0.16981437848249958400862327796, 2.18318651968304384429668611805, 3.457254114882983003224375135449, 3.98169637276613937505002239426, 4.8385095668877753191080030518, 5.79016613564941201042432410317, 6.70341434626994620871441802645, 6.973218478727710419901191979391, 8.230038662813354272604689781, 9.12690855858712117195320336383, 10.36581394696416068935182847531, 11.2870803930987770834094092540, 11.59623719455508684547395114581, 12.39258465677261805752702129966, 13.46192681142390605857518502143, 14.26244373599201417451464197326, 15.0558319010677296036013472631, 15.81619288819340103495284512108, 16.11871270718636390186413920858, 17.288496658022545496131395428880, 17.66662736949464517042664515538, 18.70307843325767532694625272849, 19.73035327931074975639213334098, 20.50634716619681986051647339450, 21.5970034227180506285427423078

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