↑

Properties

Label 1-35-35.24-r1-0-0
Degree $1$
Conductor $35$
Sign $-0.991 - 0.126i$
Analytic cond. $3.76127$
Root an. cond. $3.76127$
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  + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.5 − 0.866i)4-s − 6-s − 8-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)11-s + (−0.5 + 0.866i)12-s + 13-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (0.5 + 0.866i)18-s + (0.5 − 0.866i)19-s − 22-s + (0.5 − 0.866i)23-s + (0.5 + 0.866i)24-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.5 − 0.866i)4-s − 6-s − 8-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)11-s + (−0.5 + 0.866i)12-s + 13-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (0.5 + 0.866i)18-s + (0.5 − 0.866i)19-s − 22-s + (0.5 − 0.866i)23-s + (0.5 + 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $-0.991 - 0.126i$
Analytic conductor: \(3.76127\)
Root analytic conductor: \(3.76127\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 35,\ (1:\ ),\ -0.991 - 0.126i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08004808048 - 1.261390600i\)
\(L(\frac12)\) \(\approx\) \(0.08004808048 - 1.261390600i\)
\(L(1)\) \(\approx\) \(0.6430347134 - 0.8452580351i\)
\(L(1)\) \(\approx\) \(0.6430347134 - 0.8452580351i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 - T \)
43 \( 1 - T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (0.5 + 0.866i)T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + T \)
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

−35.829374027972310671132692815680, −34.813864953569776688268752019135, −33.50957642215133520668440025756, −33.05300824086647216780736044478, −31.695634274953445216173582936016, −30.61547307560228072304041848947, −28.79932889143834962794696440687, −27.54885927473761618127380960815, −26.3521321085271674942467350879, −25.363786422916588131252924535, −23.634173145928759461669834491444, −22.86228383194498388271602004140, −21.61138012449152128372850089441, −20.55893712554149944650185068835, −18.1535155778029157560542121697, −17.03203073071680464355718506961, −15.78986726771215064255665084717, −14.94062365075751779569511127068, −13.27148998948125456282327176985, −11.72730515797977452133779914268, −9.97260934647332206299599055075, −8.33385027943849695343667401742, −6.44518783075630195466397764088, −5.08193492197112048930689013980, −3.67908982116235042695800339759, 0.81260419161495485495030444572, 2.80905963158407783473912467722, 5.05680714376865211900938276928, 6.5122048944729669722919383988, 8.64272086683696499287044114488, 10.71694012926586080451176102810, 11.67165631871489117506271264142, 13.11104752360696230441493745778, 13.93036350819967176258809804966, 15.97369644443022968241462912703, 17.91341338700844836623097295670, 18.776034315639005751185010251, 20.055839316957369377083136194201, 21.49648215132993875939898536169, 22.7817068225103174487128289119, 23.725461334092101003273828728366, 24.85440411124297417683379062833, 26.833273852698958738955131945089, 28.39393913240658580842770388188, 29.06439858522784619202848600451, 30.283477305118937327786774839820, 31.10658707921061529622411352073, 32.50643637859832206339470019715, 33.87621702759723231891224717811, 35.26092367199733792438033029863

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