↑

Properties

Label 1-4235-4235.684-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.245 + 0.969i$
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  + (−0.969 − 0.244i)2-s + (0.669 + 0.743i)3-s + (0.879 + 0.475i)4-s + (−0.466 − 0.884i)6-s + (−0.736 − 0.676i)8-s + (−0.104 + 0.994i)9-s + (0.235 + 0.971i)12-s + (−0.610 − 0.791i)13-s + (0.548 + 0.836i)16-s + (−0.820 − 0.572i)17-s + (0.345 − 0.938i)18-s + (0.797 + 0.603i)19-s + (−0.0475 + 0.998i)23-s + (0.00951 − 0.999i)24-s + (0.398 + 0.917i)26-s + (−0.809 + 0.587i)27-s + ⋯
L(s)  = 1  + (−0.969 − 0.244i)2-s + (0.669 + 0.743i)3-s + (0.879 + 0.475i)4-s + (−0.466 − 0.884i)6-s + (−0.736 − 0.676i)8-s + (−0.104 + 0.994i)9-s + (0.235 + 0.971i)12-s + (−0.610 − 0.791i)13-s + (0.548 + 0.836i)16-s + (−0.820 − 0.572i)17-s + (0.345 − 0.938i)18-s + (0.797 + 0.603i)19-s + (−0.0475 + 0.998i)23-s + (0.00951 − 0.999i)24-s + (0.398 + 0.917i)26-s + (−0.809 + 0.587i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.245 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.245 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.245 + 0.969i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (684, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.245 + 0.969i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9743791459 + 0.7584732222i\)
\(L(\frac12)\) \(\approx\) \(0.9743791459 + 0.7584732222i\)
\(L(1)\) \(\approx\) \(0.8308908228 + 0.1979038120i\)
\(L(1)\) \(\approx\) \(0.8308908228 + 0.1979038120i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.969 - 0.244i)T \)
3 \( 1 + (0.669 + 0.743i)T \)
13 \( 1 + (-0.610 - 0.791i)T \)
17 \( 1 + (-0.820 - 0.572i)T \)
19 \( 1 + (0.797 + 0.603i)T \)
23 \( 1 + (-0.0475 + 0.998i)T \)
29 \( 1 + (-0.516 + 0.856i)T \)
31 \( 1 + (0.761 - 0.647i)T \)
37 \( 1 + (0.432 - 0.901i)T \)
41 \( 1 + (0.897 + 0.441i)T \)
43 \( 1 + (0.415 - 0.909i)T \)
47 \( 1 + (0.345 + 0.938i)T \)
53 \( 1 + (-0.548 + 0.836i)T \)
59 \( 1 + (0.0665 - 0.997i)T \)
61 \( 1 + (0.272 - 0.962i)T \)
67 \( 1 + (0.786 + 0.618i)T \)
71 \( 1 + (0.974 + 0.226i)T \)
73 \( 1 + (0.710 + 0.703i)T \)
79 \( 1 + (0.948 - 0.318i)T \)
83 \( 1 + (0.362 - 0.931i)T \)
89 \( 1 + (-0.981 - 0.189i)T \)
97 \( 1 + (-0.870 + 0.491i)T \)
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

−18.16133426197036344994594413240, −17.758683974388888884167746994385, −16.99292088896439543397323294815, −16.362882592510865871549377017907, −15.408864486194836457654126536451, −15.01965817297414326972814877125, −14.19098867752515141541914930591, −13.643678527391992013879193240683, −12.71368214235633048564931666599, −12.0040560254901376992953963616, −11.40298539291225300788513403486, −10.594802360977964618008109635488, −9.59194650291164510554270360861, −9.301214149712056665340757237877, −8.395012658688830642298288366353, −7.990756444916800915354862593135, −7.04442256118481647702638472829, −6.69153297256141148925519919648, −5.9639983077845823950541575970, −4.85894393856347376826715788927, −3.8926404217484048050146030221, −2.69735219635036466579347468196, −2.3362872214266700759164977317, −1.4283130846839997618744475062, −0.52682164381055712789472851680, 0.85320958500079752583830832417, 1.99682712824465329278577249202, 2.64916502057274224200482271970, 3.35682097369459783641020445746, 4.0948514611808117263477202256, 5.13580643333727756914690735053, 5.84665116273814206155584659398, 6.99755865779798526788641939384, 7.71800316411755968908278131210, 8.07112085557155112943639551435, 9.135600475486936263818542227944, 9.47842246545559174785109401023, 10.05361304691882109972326503992, 10.94995517244427940778271123514, 11.28889656185554408333554276441, 12.33051958732313627886294316215, 12.968299076703624004575475971491, 13.90920926689018700809997213915, 14.55644378880840015154399574607, 15.51176370457405343091377859251, 15.707805859336674343840182383965, 16.469392116886245596498032363707, 17.26098692734172604382391580891, 17.77142454094663949294631219477, 18.61571924153893381692769132222

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