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Properties

Label 1-1183-1183.10-r1-0-0
Degree $1$
Conductor $1183$
Sign $0.947 + 0.319i$
Analytic cond. $127.131$
Root an. cond. $127.131$
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  + (−0.799 + 0.600i)2-s + (0.748 + 0.663i)3-s + (0.278 − 0.960i)4-s + (−0.845 + 0.534i)5-s + (−0.996 − 0.0804i)6-s + (0.354 + 0.935i)8-s + (0.120 + 0.992i)9-s + (0.354 − 0.935i)10-s + (−0.120 + 0.992i)11-s + (0.845 − 0.534i)12-s + (−0.987 − 0.160i)15-s + (−0.845 − 0.534i)16-s + (−0.987 − 0.160i)17-s + (−0.692 − 0.721i)18-s + 19-s + (0.278 + 0.960i)20-s + ⋯
L(s)  = 1  + (−0.799 + 0.600i)2-s + (0.748 + 0.663i)3-s + (0.278 − 0.960i)4-s + (−0.845 + 0.534i)5-s + (−0.996 − 0.0804i)6-s + (0.354 + 0.935i)8-s + (0.120 + 0.992i)9-s + (0.354 − 0.935i)10-s + (−0.120 + 0.992i)11-s + (0.845 − 0.534i)12-s + (−0.987 − 0.160i)15-s + (−0.845 − 0.534i)16-s + (−0.987 − 0.160i)17-s + (−0.692 − 0.721i)18-s + 19-s + (0.278 + 0.960i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.947 + 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.947 + 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.947 + 0.319i$
Analytic conductor: \(127.131\)
Root analytic conductor: \(127.131\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (1:\ ),\ 0.947 + 0.319i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8789036297 + 0.1441087854i\)
\(L(\frac12)\) \(\approx\) \(0.8789036297 + 0.1441087854i\)
\(L(1)\) \(\approx\) \(0.6349457043 + 0.3727722410i\)
\(L(1)\) \(\approx\) \(0.6349457043 + 0.3727722410i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.799 + 0.600i)T \)
3 \( 1 + (0.748 + 0.663i)T \)
5 \( 1 + (-0.845 + 0.534i)T \)
11 \( 1 + (-0.120 + 0.992i)T \)
17 \( 1 + (-0.987 - 0.160i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.919 - 0.391i)T \)
31 \( 1 + (-0.996 - 0.0804i)T \)
37 \( 1 + (0.996 + 0.0804i)T \)
41 \( 1 + (0.948 - 0.316i)T \)
43 \( 1 + (0.428 - 0.903i)T \)
47 \( 1 + (0.692 - 0.721i)T \)
53 \( 1 + (-0.632 + 0.774i)T \)
59 \( 1 + (-0.845 + 0.534i)T \)
61 \( 1 + (0.354 - 0.935i)T \)
67 \( 1 + (0.970 + 0.239i)T \)
71 \( 1 + (0.200 - 0.979i)T \)
73 \( 1 + (-0.919 + 0.391i)T \)
79 \( 1 + (0.692 - 0.721i)T \)
83 \( 1 + (-0.748 + 0.663i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (-0.845 - 0.534i)T \)
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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

−20.58901429218944252604319181281, −20.03817760670316678065989740244, −19.53761172676547594648709730012, −18.825175494281103794609387239986, −18.14652291314724495795174128108, −17.39541788733365714832619254834, −16.25398052303540337356622626152, −15.88768567635285048752859489738, −14.82078707462932393768300147571, −13.71825228827752621532384440251, −13.00834398702707924042614589001, −12.41642249099697071622138737119, −11.366496812490009746295083069193, −11.09670317837338670208787263973, −9.49964636873268801794786048073, −9.135986717443858888959378452498, −8.18552144801392986466212680048, −7.73811577150505190251073426596, −6.95391147972587411767884300365, −5.74347409772267408989858135634, −4.21106328913230082928660822343, −3.50663525243513539706901574740, −2.70360637989822116694719665157, −1.54545069490174205012395309869, −0.75189531344123980850145487222, 0.28559952354279569797115740759, 1.94829498559802774579781698208, 2.707418077688089773169838322787, 3.99764648574531550207590410037, 4.6645951375846690818047387941, 5.78652677715241775302751185153, 7.06645998067437398410536227358, 7.48736610237020967830113747020, 8.282179670373830257072427838036, 9.18784506876674387846662939349, 9.777758583894157532162199787417, 10.72082943126885617187546007242, 11.22292837620368534325230913793, 12.39535506322870664907767379958, 13.64929519384037888216268152381, 14.4562085458070313739266498013, 15.06462901118441795920636387445, 15.64844816599318286085911485942, 16.20804665658054201307578826893, 17.128785269618115092506017752273, 18.21003868077809002176190140473, 18.62997573269783191392437889375, 19.61495079547581804773773977791, 20.256988162170713140827139690242, 20.51004882172080172988322094724

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