↑

Properties

Label 1-4235-4235.1483-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.992 - 0.123i$
Analytic cond. $19.6672$
Root an. cond. $19.6672$
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  + (−0.980 − 0.198i)2-s + (−0.951 − 0.309i)3-s + (0.921 + 0.389i)4-s + (0.870 + 0.491i)6-s + (−0.825 − 0.564i)8-s + (0.809 + 0.587i)9-s + (−0.755 − 0.654i)12-s + (−0.226 − 0.974i)13-s + (0.696 + 0.717i)16-s + (−0.931 − 0.362i)17-s + (−0.676 − 0.736i)18-s + (0.774 − 0.633i)19-s + (0.989 − 0.142i)23-s + (0.610 + 0.791i)24-s + (0.0285 + 0.999i)26-s + (−0.587 − 0.809i)27-s + ⋯
L(s)  = 1  + (−0.980 − 0.198i)2-s + (−0.951 − 0.309i)3-s + (0.921 + 0.389i)4-s + (0.870 + 0.491i)6-s + (−0.825 − 0.564i)8-s + (0.809 + 0.587i)9-s + (−0.755 − 0.654i)12-s + (−0.226 − 0.974i)13-s + (0.696 + 0.717i)16-s + (−0.931 − 0.362i)17-s + (−0.676 − 0.736i)18-s + (0.774 − 0.633i)19-s + (0.989 − 0.142i)23-s + (0.610 + 0.791i)24-s + (0.0285 + 0.999i)26-s + (−0.587 − 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.123i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.123i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.992 - 0.123i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1483, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.992 - 0.123i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03052366074 - 0.4913882872i\)
\(L(\frac12)\) \(\approx\) \(0.03052366074 - 0.4913882872i\)
\(L(1)\) \(\approx\) \(0.4643691987 - 0.1998901220i\)
\(L(1)\) \(\approx\) \(0.4643691987 - 0.1998901220i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.980 - 0.198i)T \)
3 \( 1 + (-0.951 - 0.309i)T \)
13 \( 1 + (-0.226 - 0.974i)T \)
17 \( 1 + (-0.931 - 0.362i)T \)
19 \( 1 + (0.774 - 0.633i)T \)
23 \( 1 + (0.989 - 0.142i)T \)
29 \( 1 + (0.254 - 0.967i)T \)
31 \( 1 + (-0.0855 - 0.996i)T \)
37 \( 1 + (-0.856 - 0.516i)T \)
41 \( 1 + (-0.993 + 0.113i)T \)
43 \( 1 + (0.281 - 0.959i)T \)
47 \( 1 + (0.676 - 0.736i)T \)
53 \( 1 + (0.717 + 0.696i)T \)
59 \( 1 + (0.993 + 0.113i)T \)
61 \( 1 + (-0.198 - 0.980i)T \)
67 \( 1 + (0.909 - 0.415i)T \)
71 \( 1 + (-0.998 + 0.0570i)T \)
73 \( 1 + (-0.441 - 0.897i)T \)
79 \( 1 + (-0.941 - 0.336i)T \)
83 \( 1 + (-0.884 + 0.466i)T \)
89 \( 1 + (0.841 - 0.540i)T \)
97 \( 1 + (-0.791 + 0.610i)T \)
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

−18.54876422076496947238681221972, −17.91666075989478373349414377510, −17.32408319805512309500253722284, −16.777968087781730636555411602285, −16.07744874607659373472328097566, −15.71240433795532041217726610218, −14.82112480909306645271305935046, −14.20811180015953133403969832352, −13.08474506320444827138780743384, −12.281200451127763815901956162134, −11.61741776102227402164110898188, −11.154576651049490452735873405751, −10.343006229063043494347555760973, −9.90163025055148488128811522465, −8.96308885324513677119091277526, −8.60982905897041588096879266827, −7.30287295098023829334309709135, −6.95124072890141514420493209490, −6.28130145082958837013953822982, −5.40979086089741533037519043509, −4.81292999367680326047115734464, −3.794395382908221185471761951724, −2.83298193528258958853013317918, −1.65920978096150984470609392948, −1.13104463380808116341765794877, 0.296130029458637227421516958325, 0.87227943534864282547106849790, 2.006068114945151546717496733850, 2.658954425365697845293564863532, 3.67911114979682972816047454671, 4.76383623145373792046317390067, 5.518677103860525287614805501406, 6.243991223643646520018779328966, 7.191071586131316682757481596046, 7.34221532102523512551520806505, 8.39566281380735183583749821077, 9.08744566315008840545858334129, 9.95863926033018186129739025034, 10.47334765705237592628225431767, 11.21584112014519603022822699497, 11.67965439166266677770658231974, 12.39039762967319613321425369944, 13.12737240019385031410953345772, 13.69495524364840162652691407657, 15.13567414468182848096128110317, 15.50502359139041866753406928052, 16.13813639490003895848758924277, 17.1139360270710651455362397513, 17.29533883544791096503077268994, 17.997591698973813001488000018481

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