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Properties

Label 1-116-116.63-r1-0-0
Degree $1$
Conductor $116$
Sign $-0.727 + 0.686i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.727 + 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.727 + 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(116\)    =    \(2^{2} \cdot 29\)
Sign: $-0.727 + 0.686i$
Analytic conductor: \(12.4659\)
Root analytic conductor: \(12.4659\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{116} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 116,\ (1:\ ),\ -0.727 + 0.686i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3700382744 + 0.9309512216i\)
\(L(\frac12)\) \(\approx\) \(0.3700382744 + 0.9309512216i\)
\(L(1)\) \(\approx\) \(0.7418966932 + 0.3609039609i\)
\(L(1)\) \(\approx\) \(0.7418966932 + 0.3609039609i\)

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

−28.42557725885463429179096341464, −27.80806015986273436316260926185, −27.00047639661863025054459155608, −25.06798227099674184757575058805, −24.48411453865095152537222093334, −23.69825931996011584650989854920, −22.553489342757701342313712346565, −21.477956107266114986841466990129, −20.464194851215106279630863634219, −19.17780931777788366158382801959, −18.09370690162167362187760838060, −17.153063569077814418186798308250, −16.302467353683596203040259750006, −15.10505341486239895871098366690, −13.51842914359371765458459814154, −12.566069643734215363437941179245, −11.56303213509923316826855597517, −10.71430556869625067727891474723, −8.81528334635240403417324309182, −8.036574577978951176872075268760, −6.35479402886231889162471954369, −5.33099020864045719247001942250, −4.202510078246022221902240202991, −1.81487301258962716689771888051, −0.48540657349548177258942072483, 1.68532949907392948284878133972, 3.82644700916630757462806847640, 4.766657294874706325755261119307, 6.4438696631553574131692179394, 7.18585860245597687574030948736, 8.967345198660467885533812466486, 10.41617285761140462749461685622, 11.16289680699565123375085437958, 11.97599408588127071195967049795, 13.68422564502889929538093955242, 14.88095297698813281786551597013, 15.661432922315774937483899478924, 17.17440483303410130404001597469, 17.72557048074977710961937979413, 18.86557639296722033663144860108, 20.23231946800144910196048349696, 21.45044519415071394803074774053, 22.19898880395788810549936443229, 23.338822602805868809382400450679, 23.84664692861031135161498832201, 25.48163918126750543435362149264, 26.612027842415981944958586460360, 27.35437704281223397725874832375, 28.21533447540567617889228326195, 29.39630236879131770500742869289

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