↑

Properties

Label 1-115-115.82-r1-0-0
Degree $1$
Conductor $115$
Sign $-0.148 - 0.988i$
Analytic cond. $12.3584$
Root an. cond. $12.3584$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.989 − 0.142i)2-s + (0.909 − 0.415i)3-s + (0.959 + 0.281i)4-s + (−0.959 + 0.281i)6-s + (−0.540 + 0.841i)7-s + (−0.909 − 0.415i)8-s + (0.654 − 0.755i)9-s + (−0.142 − 0.989i)11-s + (0.989 − 0.142i)12-s + (−0.540 − 0.841i)13-s + (0.654 − 0.755i)14-s + (0.841 + 0.540i)16-s + (−0.281 − 0.959i)17-s + (−0.755 + 0.654i)18-s + (0.959 + 0.281i)19-s + ⋯
L(s)  = 1  + (−0.989 − 0.142i)2-s + (0.909 − 0.415i)3-s + (0.959 + 0.281i)4-s + (−0.959 + 0.281i)6-s + (−0.540 + 0.841i)7-s + (−0.909 − 0.415i)8-s + (0.654 − 0.755i)9-s + (−0.142 − 0.989i)11-s + (0.989 − 0.142i)12-s + (−0.540 − 0.841i)13-s + (0.654 − 0.755i)14-s + (0.841 + 0.540i)16-s + (−0.281 − 0.959i)17-s + (−0.755 + 0.654i)18-s + (0.959 + 0.281i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.148 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.148 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $-0.148 - 0.988i$
Analytic conductor: \(12.3584\)
Root analytic conductor: \(12.3584\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 115,\ (1:\ ),\ -0.148 - 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7990527392 - 0.9276225821i\)
\(L(\frac12)\) \(\approx\) \(0.7990527392 - 0.9276225821i\)
\(L(1)\) \(\approx\) \(0.8365062857 - 0.3136438592i\)
\(L(1)\) \(\approx\) \(0.8365062857 - 0.3136438592i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.989 - 0.142i)T \)
3 \( 1 + (0.909 - 0.415i)T \)
7 \( 1 + (-0.540 + 0.841i)T \)
11 \( 1 + (-0.142 - 0.989i)T \)
13 \( 1 + (-0.540 - 0.841i)T \)
17 \( 1 + (-0.281 - 0.959i)T \)
19 \( 1 + (0.959 + 0.281i)T \)
29 \( 1 + (0.959 - 0.281i)T \)
31 \( 1 + (0.415 - 0.909i)T \)
37 \( 1 + (-0.755 - 0.654i)T \)
41 \( 1 + (-0.654 - 0.755i)T \)
43 \( 1 + (0.909 - 0.415i)T \)
47 \( 1 + iT \)
53 \( 1 + (0.540 - 0.841i)T \)
59 \( 1 + (-0.841 + 0.540i)T \)
61 \( 1 + (0.415 - 0.909i)T \)
67 \( 1 + (-0.989 - 0.142i)T \)
71 \( 1 + (-0.142 + 0.989i)T \)
73 \( 1 + (-0.281 + 0.959i)T \)
79 \( 1 + (-0.841 + 0.540i)T \)
83 \( 1 + (0.755 + 0.654i)T \)
89 \( 1 + (-0.415 - 0.909i)T \)
97 \( 1 + (0.755 - 0.654i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−29.08042149019699846785425089337, −28.18851055847514767704467375080, −26.93480220182670360686102005161, −26.349750325416301964523429786628, −25.62667563809335711105527961677, −24.56439674037516679018376627783, −23.44973905818611414772147315572, −21.8666008784983244570745457966, −20.69643523265358530317812996294, −19.8340084976598819991274765949, −19.27257914065937270338241373398, −17.86949386206493086061666978173, −16.759627434818229517532256007478, −15.808437137727547553474725092202, −14.81991628259229007730022352226, −13.67974497006110284409709012147, −12.18473604331830419601394725764, −10.514853510679737873894955678, −9.85763303235139439962887648759, −8.84399966980827804211480269866, −7.56303391490992144376908368317, −6.739824535543346572063939318466, −4.60189885016039963845304811490, −3.055725956546618373764644456787, −1.60602442761718744778100138360, 0.6329663120076889410963606240, 2.454080049940352933651059639321, 3.237806233321241311013188650798, 5.80104159523875976737970232319, 7.1569460115935142403332398375, 8.22444432778016374636722109266, 9.14457284062025235767036481147, 10.080031324064784005174997131280, 11.68273525088371569338371927657, 12.654517623223872347949154175074, 13.96550244106509578831783002716, 15.41249816579287748760588432451, 16.04963428340393859588033162172, 17.64285307979456347731911245901, 18.60272287127820038009987230978, 19.271303246619135785269580079292, 20.24904997292474316873273796688, 21.20600041299402668636562850255, 22.4315243474528438997030533707, 24.354262727355115167582945829, 24.812764860008161404536313214671, 25.756521769865358291645977177087, 26.72860039988731375572275145727, 27.4969713603134548466249058908, 28.98265037456341594713682905657

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