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Properties

Label 1-117-117.68-r1-0-0
Degree $1$
Conductor $117$
Sign $-0.822 + 0.568i$
Analytic cond. $12.5733$
Root an. cond. $12.5733$
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  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)4-s + (0.5 + 0.866i)5-s + 7-s − 8-s + (−0.5 + 0.866i)10-s + (0.5 + 0.866i)11-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.5 − 0.866i)19-s − 20-s + (−0.5 + 0.866i)22-s − 23-s + (−0.5 + 0.866i)25-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)4-s + (0.5 + 0.866i)5-s + 7-s − 8-s + (−0.5 + 0.866i)10-s + (0.5 + 0.866i)11-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.5 − 0.866i)19-s − 20-s + (−0.5 + 0.866i)22-s − 23-s + (−0.5 + 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.822 + 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.822 + 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.822 + 0.568i$
Analytic conductor: \(12.5733\)
Root analytic conductor: \(12.5733\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 117,\ (1:\ ),\ -0.822 + 0.568i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7129407882 + 2.286695855i\)
\(L(\frac12)\) \(\approx\) \(0.7129407882 + 2.286695855i\)
\(L(1)\) \(\approx\) \(1.096659396 + 1.076705519i\)
\(L(1)\) \(\approx\) \(1.096659396 + 1.076705519i\)

Euler product

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

−28.690228464143363169257903547858, −27.63781904575283251301711043784, −27.04787445565324626969044797391, −25.15149451210424173036377489248, −24.33102042356988870524766674309, −23.487591785229379769414977043518, −22.14820148292029303747906767696, −21.16898402718129418659676303129, −20.648837045723227283030462403625, −19.48386938818222555178153918756, −18.30942917193152409073883014635, −17.27450487586651873231109169712, −15.984912698925450332150479916993, −14.31421129622169739769578169036, −13.8710777715597910034516860558, −12.46299430008337796405156254338, −11.66036544851526430739467944584, −10.44303937359024585758281461746, −9.22587997500589986919847814364, −8.19140142756109404721317611583, −6.00869831716088536228398443565, −5.053298388428911807404992510891, −3.86204776686037631673311859052, −2.082140981527442944547507065430, −0.896283431758048024450220189375, 2.09168494773958843464213328746, 3.805198789721316099371609603616, 5.08530444768906359369326464298, 6.35427329572045793403042395584, 7.34509480243945881993712919491, 8.53299761513725580642184530374, 9.97977684593401650621711763321, 11.38028040527840996772112112951, 12.616539990855782549522804086066, 13.95609281332872886134324898046, 14.66311315857482688167948325269, 15.43724072065844049984297569126, 17.111708550519466151840630862558, 17.64437552130314315341990283844, 18.68296088035400766994213177288, 20.37094430570287606869949446693, 21.60866120986014448305643569539, 22.17811286119250912288072070815, 23.40930781330310560360885499949, 24.1987626294900211503147220034, 25.46212633330155922249362771242, 25.96625645477398281325685984229, 27.169469468584035413028434625518, 28.07811461548371715883497517128, 29.888258407107738089165665577823

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