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Properties

Label 1-117-117.43-r0-0-0
Degree $1$
Conductor $117$
Sign $-0.632 - 0.774i$
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  + (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

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.632 - 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.632 - 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.632 - 0.774i$
Analytic conductor: \(0.543345\)
Root analytic conductor: \(0.543345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 117,\ (0:\ ),\ -0.632 - 0.774i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5133847946 - 1.082776824i\)
\(L(\frac12)\) \(\approx\) \(0.5133847946 - 1.082776824i\)
\(L(1)\) \(\approx\) \(0.8992566299 - 0.8021354864i\)
\(L(1)\) \(\approx\) \(0.8992566299 - 0.8021354864i\)

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

−29.74889252836870436478738167105, −28.723108435843472647071125966604, −27.093564499357446349047925158581, −26.36951457423706880784008557742, −25.293492619586797765836342421597, −24.88255044392505617201324482417, −23.06331872842921790767918205337, −22.73629611696210084769444637202, −21.8049920758952539598225119587, −20.54273759263572342131098251773, −19.0269601911857548956384056061, −17.9936916613535204208820166707, −17.02245151093483786764931893981, −15.87232994472079776199464107364, −14.90058288060607966757068453464, −13.91919747606064316693075193977, −12.957922256878578286809340880208, −11.751391093623702975190749384155, −10.02920338425260645141850358093, −9.109937631951226384433501091875, −7.30562172020691938545762887900, −6.65826715129526964873173921405, −5.488518560206830654438363569458, −3.88797744415610632744431664540, −2.65339572180239406505213331987, 1.06726728616764591500577807282, 2.7703462712490531477065335756, 4.08391149795059537714358061855, 5.45190809315691215179970581366, 6.47984808271072637296876658778, 8.77136883884869914270007044506, 9.46302756687293528742058201169, 10.72547359364487813723381010591, 11.9583761565306405238062909991, 13.113144680860036222758287628720, 13.5488898392582044145757642241, 15.062782258058054834523008866025, 16.35299146702133881713078400949, 17.452665040737045737341282009914, 18.90282786874452226893824271102, 19.70288910552578613602715925928, 20.6309449556168176383265118254, 21.78116924960697140747742352977, 22.35420644028214165359405591171, 23.77079086727197067956181802948, 24.49352206628938242808213378310, 25.78236664398388252071522692189, 27.07021146914976658717397360601, 28.22786696568445150120415558383, 28.94958873809557974048010987787

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