Properties

Label 1-4235-4235.2609-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.161 - 0.986i$
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  + (−0.683 − 0.730i)2-s + (0.669 + 0.743i)3-s + (−0.0665 + 0.997i)4-s + (0.0855 − 0.996i)6-s + (0.774 − 0.633i)8-s + (−0.104 + 0.994i)9-s + (−0.786 + 0.618i)12-s + (0.466 − 0.884i)13-s + (−0.991 − 0.132i)16-s + (0.948 + 0.318i)17-s + (0.797 − 0.603i)18-s + (−0.595 − 0.803i)19-s + (−0.723 − 0.690i)23-s + (0.988 + 0.151i)24-s + (−0.964 + 0.263i)26-s + (−0.809 + 0.587i)27-s + ⋯
L(s)  = 1  + (−0.683 − 0.730i)2-s + (0.669 + 0.743i)3-s + (−0.0665 + 0.997i)4-s + (0.0855 − 0.996i)6-s + (0.774 − 0.633i)8-s + (−0.104 + 0.994i)9-s + (−0.786 + 0.618i)12-s + (0.466 − 0.884i)13-s + (−0.991 − 0.132i)16-s + (0.948 + 0.318i)17-s + (0.797 − 0.603i)18-s + (−0.595 − 0.803i)19-s + (−0.723 − 0.690i)23-s + (0.988 + 0.151i)24-s + (−0.964 + 0.263i)26-s + (−0.809 + 0.587i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.161 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.161 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.161 - 0.986i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (2609, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.161 - 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9709490241 - 0.8248865199i\)
\(L(\frac12)\) \(\approx\) \(0.9709490241 - 0.8248865199i\)
\(L(1)\) \(\approx\) \(0.9162658116 - 0.1617290803i\)
\(L(1)\) \(\approx\) \(0.9162658116 - 0.1617290803i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.683 + 0.730i)T \)
3 \( 1 + (-0.669 - 0.743i)T \)
13 \( 1 + (-0.466 + 0.884i)T \)
17 \( 1 + (-0.948 - 0.318i)T \)
19 \( 1 + (0.595 + 0.803i)T \)
23 \( 1 + (0.723 + 0.690i)T \)
29 \( 1 + (-0.736 + 0.676i)T \)
31 \( 1 + (0.272 + 0.962i)T \)
37 \( 1 + (0.640 + 0.768i)T \)
41 \( 1 + (-0.516 - 0.856i)T \)
43 \( 1 + (-0.841 - 0.540i)T \)
47 \( 1 + (-0.797 - 0.603i)T \)
53 \( 1 + (-0.991 + 0.132i)T \)
59 \( 1 + (0.483 + 0.875i)T \)
61 \( 1 + (0.290 + 0.956i)T \)
67 \( 1 + (-0.327 - 0.945i)T \)
71 \( 1 + (0.870 + 0.491i)T \)
73 \( 1 + (0.997 - 0.0760i)T \)
79 \( 1 + (0.449 + 0.893i)T \)
83 \( 1 + (0.941 - 0.336i)T \)
89 \( 1 + (-0.995 + 0.0950i)T \)
97 \( 1 + (0.362 + 0.931i)T \)
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

−18.62572907574470211938380541703, −17.89973634223597797308110909278, −17.28104409774322848816321838446, −16.46684559035325676199928827690, −15.92585989101209671188895744919, −15.123538298379306953017126473701, −14.38096868251227035970545600933, −13.97222022420700024789392125546, −13.431068972911418254971362631903, −12.22272531178356867558492445867, −11.9448779038651589563351438752, −10.73971884225251682203711602735, −10.135763737805679371395902018859, −9.289784980778632491123350829369, −8.69888558632851657654992836793, −8.20617453875124236805569917126, −7.270744464579171811162041201949, −6.99905026163234541296277420992, −6.015310788296750689695055915192, −5.54872928556963541270635303055, −4.32966108357903220791429979866, −3.55687192448868658722442067652, −2.47870902502200893356618743249, −1.60234550486114623923733128295, −1.08126671958128197784945700027, 0.43307377789925240623536094957, 1.55617439719904064832355204753, 2.55654524959019800905852277644, 2.94986518191603734265131971335, 3.97083305325856181175729239609, 4.334066311182274339211787645600, 5.43798219225704391295260319084, 6.33863865379464601436629829932, 7.56393100610804961806980236167, 7.97749889026771428359453684570, 8.65124659257985388136334003931, 9.306451988250540433361494762506, 10.02335749587210190683912969134, 10.55636942485518668341268248152, 11.08692171041059909761762391929, 11.9798591454454275787043188228, 12.80552994278267097307030711130, 13.30886148481030474341690114220, 14.17641197990791398891004590140, 14.83696943295567572324932650857, 15.774634954416953623019000874651, 16.08560133569491878633245019816, 17.017559755662680100655875020235, 17.51724811453723849231834975602, 18.3547885001786308599354946841

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