↑

Properties

Label 1-4235-4235.3587-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.718 - 0.695i$
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  + (−0.371 + 0.928i)2-s + (0.866 − 0.5i)3-s + (−0.723 − 0.690i)4-s + (0.142 + 0.989i)6-s + (0.909 − 0.415i)8-s + (0.5 − 0.866i)9-s + (−0.971 − 0.235i)12-s + (−0.281 − 0.959i)13-s + (0.0475 + 0.998i)16-s + (0.945 + 0.327i)17-s + (0.618 + 0.786i)18-s + (−0.327 − 0.945i)19-s + (0.998 − 0.0475i)23-s + (0.580 − 0.814i)24-s + (0.995 + 0.0950i)26-s − i·27-s + ⋯
L(s)  = 1  + (−0.371 + 0.928i)2-s + (0.866 − 0.5i)3-s + (−0.723 − 0.690i)4-s + (0.142 + 0.989i)6-s + (0.909 − 0.415i)8-s + (0.5 − 0.866i)9-s + (−0.971 − 0.235i)12-s + (−0.281 − 0.959i)13-s + (0.0475 + 0.998i)16-s + (0.945 + 0.327i)17-s + (0.618 + 0.786i)18-s + (−0.327 − 0.945i)19-s + (0.998 − 0.0475i)23-s + (0.580 − 0.814i)24-s + (0.995 + 0.0950i)26-s − i·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.718 - 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.718 - 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.718 - 0.695i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (3587, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.718 - 0.695i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.781472604 - 0.7211746516i\)
\(L(\frac12)\) \(\approx\) \(1.781472604 - 0.7211746516i\)
\(L(1)\) \(\approx\) \(1.183466963 + 0.02139792740i\)
\(L(1)\) \(\approx\) \(1.183466963 + 0.02139792740i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.371 + 0.928i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (-0.281 - 0.959i)T \)
17 \( 1 + (0.945 + 0.327i)T \)
19 \( 1 + (-0.327 - 0.945i)T \)
23 \( 1 + (0.998 - 0.0475i)T \)
29 \( 1 + (0.654 - 0.755i)T \)
31 \( 1 + (-0.235 - 0.971i)T \)
37 \( 1 + (0.690 + 0.723i)T \)
41 \( 1 + (0.142 + 0.989i)T \)
43 \( 1 + (0.909 - 0.415i)T \)
47 \( 1 + (-0.618 + 0.786i)T \)
53 \( 1 + (-0.998 - 0.0475i)T \)
59 \( 1 + (0.928 - 0.371i)T \)
61 \( 1 + (0.786 + 0.618i)T \)
67 \( 1 + (-0.618 - 0.786i)T \)
71 \( 1 + (-0.654 + 0.755i)T \)
73 \( 1 + (0.458 - 0.888i)T \)
79 \( 1 + (-0.580 - 0.814i)T \)
83 \( 1 + (-0.540 + 0.841i)T \)
89 \( 1 + (0.981 - 0.189i)T \)
97 \( 1 + (0.909 - 0.415i)T \)
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

−18.87138765031620424469269248948, −17.94813991443422307062982110925, −17.12463991614464529994422270864, −16.334018625223267912862087129426, −16.06602749131421600775765265741, −14.69944726938966847881890560844, −14.389176090730376137104283950770, −13.7879084631023308446165087982, −12.84699153998630142939504650559, −12.38960346391403237573329306212, −11.5209468600090444788392885623, −10.76874791307036658962577703692, −10.15933454110948788893747491413, −9.52578534917889530338320304739, −8.92185554750828536535741890379, −8.34827441968869333607777330373, −7.523386811828232338114747691167, −6.920868244858600509198386547422, −5.49617318144007001610665807139, −4.76863349552728768373171715425, −4.029661278275962880890171772725, −3.37024186483627395111728848899, −2.68141352612167275106782945328, −1.86616664010506746958971760256, −1.12018858952136121529102127230, 0.61384823538631081471251915085, 1.30705248614011106446522830637, 2.48397125106006174730331115377, 3.1806187483684476041741068563, 4.20869850134782796918225518366, 4.94046820092051597907982484647, 5.895161401300800021709035669412, 6.50141250352274216114310688229, 7.32380263669036502305098927575, 7.90128135179028762369341432559, 8.347857517998592483409756066069, 9.240679283930377406084524945849, 9.73268760706897236701929025895, 10.46765416775080715190135089071, 11.40748197304022958986058941486, 12.490670146295032190189745986976, 13.13177018742663395242311361098, 13.502365602974612780648441320959, 14.58410518746072839697373645785, 14.794649262582428265959107007728, 15.47167325968395215617809293368, 16.15694389914196945916486370445, 17.13652858503767525474099167657, 17.53785873450734441180286555912, 18.236951588293594305603105702585

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