Properties

Label 1-4235-4235.2923-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.176 - 0.984i$
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  + (0.524 − 0.851i)2-s + (−0.406 + 0.913i)3-s + (−0.449 − 0.893i)4-s + (0.564 + 0.825i)6-s + (−0.996 − 0.0855i)8-s + (−0.669 − 0.743i)9-s + (0.998 − 0.0475i)12-s + (−0.967 + 0.254i)13-s + (−0.595 + 0.803i)16-s + (−0.768 + 0.640i)17-s + (−0.983 + 0.179i)18-s + (−0.969 + 0.244i)19-s + (−0.814 + 0.580i)23-s + (0.483 − 0.875i)24-s + (−0.290 + 0.956i)26-s + (0.951 − 0.309i)27-s + ⋯
L(s)  = 1  + (0.524 − 0.851i)2-s + (−0.406 + 0.913i)3-s + (−0.449 − 0.893i)4-s + (0.564 + 0.825i)6-s + (−0.996 − 0.0855i)8-s + (−0.669 − 0.743i)9-s + (0.998 − 0.0475i)12-s + (−0.967 + 0.254i)13-s + (−0.595 + 0.803i)16-s + (−0.768 + 0.640i)17-s + (−0.983 + 0.179i)18-s + (−0.969 + 0.244i)19-s + (−0.814 + 0.580i)23-s + (0.483 − 0.875i)24-s + (−0.290 + 0.956i)26-s + (0.951 − 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.176 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.176 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.176 - 0.984i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (2923, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.176 - 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6015236119 - 0.5032011009i\)
\(L(\frac12)\) \(\approx\) \(0.6015236119 - 0.5032011009i\)
\(L(1)\) \(\approx\) \(0.8213380517 - 0.1807334403i\)
\(L(1)\) \(\approx\) \(0.8213380517 - 0.1807334403i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.524 - 0.851i)T \)
3 \( 1 + (-0.406 + 0.913i)T \)
13 \( 1 + (-0.967 + 0.254i)T \)
17 \( 1 + (-0.768 + 0.640i)T \)
19 \( 1 + (-0.969 + 0.244i)T \)
23 \( 1 + (-0.814 + 0.580i)T \)
29 \( 1 + (-0.466 + 0.884i)T \)
31 \( 1 + (-0.935 - 0.353i)T \)
37 \( 1 + (0.151 - 0.988i)T \)
41 \( 1 + (-0.610 + 0.791i)T \)
43 \( 1 + (0.755 + 0.654i)T \)
47 \( 1 + (0.983 + 0.179i)T \)
53 \( 1 + (0.803 - 0.595i)T \)
59 \( 1 + (-0.380 + 0.924i)T \)
61 \( 1 + (-0.879 - 0.475i)T \)
67 \( 1 + (0.690 - 0.723i)T \)
71 \( 1 + (0.897 - 0.441i)T \)
73 \( 1 + (0.508 - 0.861i)T \)
79 \( 1 + (0.123 - 0.992i)T \)
83 \( 1 + (0.676 + 0.736i)T \)
89 \( 1 + (0.786 + 0.618i)T \)
97 \( 1 + (0.856 - 0.516i)T \)
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

−18.55353944606293014842589530787, −17.50537283482565150919162993052, −17.27322034290034472099391601915, −16.61983132444080004993899046855, −15.78057916529503247170842595897, −15.13258961356504375370787108486, −14.382805395356351101220509051466, −13.74071620289451361079668467988, −13.18514218707264314455776848371, −12.42156564310113023064547212076, −12.05826314577738989015474102918, −11.21455534074445740967422502505, −10.38211835170084230172927483128, −9.31018046325733558769709880386, −8.584736961868993596405518588670, −7.89848488040034566379572212976, −7.194142458157772403222084376574, −6.72010451050542094846209227059, −5.96269718646258446849145268688, −5.27826333795465906701966747933, −4.59168083154165279697657045379, −3.770520683212058930059433002441, −2.49827178359752120089589321414, −2.21486514333175736255671546561, −0.58695146507317269824203206184, 0.299955785036943222049515879118, 1.733432322891633325999831587961, 2.37345025067416571198839165921, 3.40233086155978085495997974647, 4.05376957082539804067809920137, 4.58276480434884331219366540215, 5.39335288330193893771304908351, 6.00987979870152452738414161744, 6.7634355447341788429182737263, 7.93365794599581146755004479392, 9.00065878136792996110079701126, 9.3647969202730000431497428860, 10.145461633462749946816608756167, 10.8592042108596342832434729003, 11.1531251122117961713092878081, 12.21256844846135244484302953975, 12.46855704276469232951952136092, 13.417950197065707140823074344243, 14.221758418454807897057957449377, 14.95843138193135294654316544826, 15.156237809700979690150386120152, 16.20970517603043041066739041746, 16.82650287839952987548457305269, 17.63714845345910881589315325799, 18.159450203136213172486027506813

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