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Properties

Label 1-116-116.111-r1-0-0
Degree $1$
Conductor $116$
Sign $0.990 - 0.136i$
Analytic cond. $12.4659$
Root an. cond. $12.4659$
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  + (0.900 − 0.433i)3-s + (−0.222 + 0.974i)5-s + (0.900 − 0.433i)7-s + (0.623 − 0.781i)9-s + (−0.623 − 0.781i)11-s + (0.623 + 0.781i)13-s + (0.222 + 0.974i)15-s + 17-s + (0.900 + 0.433i)19-s + (0.623 − 0.781i)21-s + (0.222 + 0.974i)23-s + (−0.900 − 0.433i)25-s + (0.222 − 0.974i)27-s + (0.222 − 0.974i)31-s + (−0.900 − 0.433i)33-s + ⋯
L(s)  = 1  + (0.900 − 0.433i)3-s + (−0.222 + 0.974i)5-s + (0.900 − 0.433i)7-s + (0.623 − 0.781i)9-s + (−0.623 − 0.781i)11-s + (0.623 + 0.781i)13-s + (0.222 + 0.974i)15-s + 17-s + (0.900 + 0.433i)19-s + (0.623 − 0.781i)21-s + (0.222 + 0.974i)23-s + (−0.900 − 0.433i)25-s + (0.222 − 0.974i)27-s + (0.222 − 0.974i)31-s + (−0.900 − 0.433i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.990 - 0.136i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.990 - 0.136i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(116\)    =    \(2^{2} \cdot 29\)
Sign: $0.990 - 0.136i$
Analytic conductor: \(12.4659\)
Root analytic conductor: \(12.4659\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{116} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 116,\ (1:\ ),\ 0.990 - 0.136i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.605722684 - 0.1788322563i\)
\(L(\frac12)\) \(\approx\) \(2.605722684 - 0.1788322563i\)
\(L(1)\) \(\approx\) \(1.606695673 - 0.07978076503i\)
\(L(1)\) \(\approx\) \(1.606695673 - 0.07978076503i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
29 \( 1 \)
good3 \( 1 + (0.900 - 0.433i)T \)
5 \( 1 + (-0.222 + 0.974i)T \)
7 \( 1 + (0.900 - 0.433i)T \)
11 \( 1 + (-0.623 - 0.781i)T \)
13 \( 1 + (0.623 + 0.781i)T \)
17 \( 1 + T \)
19 \( 1 + (0.900 + 0.433i)T \)
23 \( 1 + (0.222 + 0.974i)T \)
31 \( 1 + (0.222 - 0.974i)T \)
37 \( 1 + (0.623 - 0.781i)T \)
41 \( 1 + T \)
43 \( 1 + (0.222 + 0.974i)T \)
47 \( 1 + (-0.623 - 0.781i)T \)
53 \( 1 + (-0.222 + 0.974i)T \)
59 \( 1 - T \)
61 \( 1 + (-0.900 + 0.433i)T \)
67 \( 1 + (-0.623 + 0.781i)T \)
71 \( 1 + (-0.623 - 0.781i)T \)
73 \( 1 + (-0.222 - 0.974i)T \)
79 \( 1 + (-0.623 + 0.781i)T \)
83 \( 1 + (0.900 + 0.433i)T \)
89 \( 1 + (-0.222 + 0.974i)T \)
97 \( 1 + (-0.900 - 0.433i)T \)
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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

−28.712123682058487098620191746760, −27.85026318392769533059726166878, −27.17536189674638466051355360960, −25.846936440802815714041584887987, −24.99644793699384298430595799763, −24.18818103789480218104094982351, −22.95371278520664817706521574511, −21.45326280440632041082289558671, −20.65133624022906279534696198515, −20.16169471870365094238573260634, −18.72842540676859277404234868985, −17.6483857037064474281564180164, −16.19729883744089880569359368444, −15.41557845129034589277291575647, −14.396596150297252728785922608722, −13.159437891699457096910647440939, −12.163098610106171220559300819485, −10.64421336156950651957617007775, −9.420787397744016831169448383900, −8.33334314389066684851669253390, −7.668467476676202530479186091981, −5.34134751291679794546364707188, −4.52297500713767677812832313131, −2.93396509495398010884815343621, −1.36220418321631452417565698090, 1.34183773188298034020538857351, 2.894113142654038610669758214010, 3.97106430424402803736704815534, 5.93497380537523229369833880916, 7.457372843428398700510945476387, 7.97320704499929586777525812216, 9.48148435598963105638437414501, 10.83812106040269237587151758073, 11.81128026893615998966515845990, 13.52238331462595726638897456031, 14.1360001878777962196985769402, 15.05004637278292052772067766965, 16.310365561103356939558371959758, 17.97743766313745069675738031333, 18.633798598996926263906967483414, 19.53417382673814358877038916668, 20.86217559669217464737434054870, 21.468329254215089501368918775675, 23.16027043607564266345879270347, 23.85662042841250140452397258320, 24.93855491081815693757611468656, 26.19161001431328037167145486013, 26.606952237124626767949868296556, 27.71073841277574817583524892652, 29.398959414991892374681963206256

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