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Properties

Label 1-117-117.32-r0-0-0
Degree $1$
Conductor $117$
Sign $0.790 + 0.612i$
Analytic cond. $0.543345$
Root an. cond. $0.543345$
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  + i·2-s − 4-s + (0.866 − 0.5i)5-s + (0.866 − 0.5i)7-s − i·8-s + (0.5 + 0.866i)10-s − i·11-s + (0.5 + 0.866i)14-s + 16-s + (−0.5 + 0.866i)17-s + (0.866 + 0.5i)19-s + (−0.866 + 0.5i)20-s + 22-s + (−0.5 + 0.866i)23-s + (0.5 − 0.866i)25-s + ⋯
L(s)  = 1  + i·2-s − 4-s + (0.866 − 0.5i)5-s + (0.866 − 0.5i)7-s − i·8-s + (0.5 + 0.866i)10-s − i·11-s + (0.5 + 0.866i)14-s + 16-s + (−0.5 + 0.866i)17-s + (0.866 + 0.5i)19-s + (−0.866 + 0.5i)20-s + 22-s + (−0.5 + 0.866i)23-s + (0.5 − 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.790 + 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.790 + 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.790 + 0.612i$
Analytic conductor: \(0.543345\)
Root analytic conductor: \(0.543345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 117,\ (0:\ ),\ 0.790 + 0.612i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.084473003 + 0.3711669437i\)
\(L(\frac12)\) \(\approx\) \(1.084473003 + 0.3711669437i\)
\(L(1)\) \(\approx\) \(1.076342872 + 0.3408161001i\)
\(L(1)\) \(\approx\) \(1.076342872 + 0.3408161001i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + iT \)
5 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (0.866 - 0.5i)T \)
11 \( 1 - iT \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 - T \)
31 \( 1 + (-0.866 + 0.5i)T \)
37 \( 1 + (0.866 - 0.5i)T \)
41 \( 1 + (-0.866 - 0.5i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (0.866 + 0.5i)T \)
53 \( 1 - T \)
59 \( 1 - iT \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 + (-0.866 - 0.5i)T \)
73 \( 1 + iT \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.866 - 0.5i)T \)
89 \( 1 + (-0.866 + 0.5i)T \)
97 \( 1 + (-0.866 + 0.5i)T \)
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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

−29.00638543820103439760457076344, −28.335924752871558831307593183, −27.25421131253605782464790091934, −26.2598147210886703216912215776, −25.14048625040721397826063713723, −23.92807844382559573578896103306, −22.44662578474559876147541876144, −22.0265304595069279830151089080, −20.75576282652860205955311093764, −20.197040217544764589167276855181, −18.42026194063503806035385050392, −18.1683461349890375161925429845, −17.07227836726031154488466118851, −15.12133114046171197940876526904, −14.206551862047991776410736285854, −13.19753979939308341784885156213, −11.94088380636840805802608765193, −10.99531774494956403601307344380, −9.83156438378027846512247283610, −8.96143976581679807181882121777, −7.37630075588936149173714928490, −5.59263754634733215081384634264, −4.523757846112522164592665129414, −2.7026073621246128962071684362, −1.77484582237474436167410742791, 1.42460342153674582930028982875, 3.85016766812350400478500376694, 5.235679099925328721713689948, 6.06064364962279626518454535026, 7.59019241305793523599444156286, 8.59121596865446342885394040992, 9.68053449349857158265275047178, 11.04033467186475475420091252014, 12.78891149916062185124377538018, 13.79720737672547864762806805664, 14.45495915099072574933183669243, 15.90375393177197575573727752855, 16.89778557157486760793793002270, 17.62108790140850655097240748477, 18.60489302842268185222763023838, 20.13846459221768457470962468338, 21.40295520549214988565757814530, 22.11675212175813659153167967910, 23.68037848582442452856606983023, 24.23679348092300826283085663468, 25.12202629358778615778602552641, 26.22826001300259929428539009291, 27.07474434947561258810727555256, 28.09346633259996959586739414707, 29.26697015473543578176123693656

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