Properties

Label 1-4235-4235.2938-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.662 - 0.749i$
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  + (0.690 − 0.723i)2-s + (0.866 + 0.5i)3-s + (−0.0475 − 0.998i)4-s + (0.959 − 0.281i)6-s + (−0.755 − 0.654i)8-s + (0.5 + 0.866i)9-s + (0.458 − 0.888i)12-s + (−0.540 − 0.841i)13-s + (−0.995 + 0.0950i)16-s + (0.618 − 0.786i)17-s + (0.971 + 0.235i)18-s + (−0.786 + 0.618i)19-s + (−0.0950 − 0.995i)23-s + (−0.327 − 0.945i)24-s + (−0.981 − 0.189i)26-s + i·27-s + ⋯
L(s)  = 1  + (0.690 − 0.723i)2-s + (0.866 + 0.5i)3-s + (−0.0475 − 0.998i)4-s + (0.959 − 0.281i)6-s + (−0.755 − 0.654i)8-s + (0.5 + 0.866i)9-s + (0.458 − 0.888i)12-s + (−0.540 − 0.841i)13-s + (−0.995 + 0.0950i)16-s + (0.618 − 0.786i)17-s + (0.971 + 0.235i)18-s + (−0.786 + 0.618i)19-s + (−0.0950 − 0.995i)23-s + (−0.327 − 0.945i)24-s + (−0.981 − 0.189i)26-s + i·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.662 - 0.749i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (2938, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.662 - 0.749i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.126787176 - 2.498991195i\)
\(L(\frac12)\) \(\approx\) \(1.126787176 - 2.498991195i\)
\(L(1)\) \(\approx\) \(1.579226618 - 0.8191173090i\)
\(L(1)\) \(\approx\) \(1.579226618 - 0.8191173090i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.690 - 0.723i)T \)
3 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (-0.540 - 0.841i)T \)
17 \( 1 + (0.618 - 0.786i)T \)
19 \( 1 + (-0.786 + 0.618i)T \)
23 \( 1 + (-0.0950 - 0.995i)T \)
29 \( 1 + (0.142 + 0.989i)T \)
31 \( 1 + (0.888 - 0.458i)T \)
37 \( 1 + (-0.998 - 0.0475i)T \)
41 \( 1 + (0.959 - 0.281i)T \)
43 \( 1 + (-0.755 - 0.654i)T \)
47 \( 1 + (-0.971 + 0.235i)T \)
53 \( 1 + (0.0950 - 0.995i)T \)
59 \( 1 + (0.723 - 0.690i)T \)
61 \( 1 + (-0.235 - 0.971i)T \)
67 \( 1 + (-0.971 - 0.235i)T \)
71 \( 1 + (-0.142 - 0.989i)T \)
73 \( 1 + (0.814 - 0.580i)T \)
79 \( 1 + (0.327 - 0.945i)T \)
83 \( 1 + (0.909 - 0.415i)T \)
89 \( 1 + (0.928 - 0.371i)T \)
97 \( 1 + (-0.755 - 0.654i)T \)
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

−18.658922369740693617286074657813, −17.709547316261341012011830455107, −17.31353434529088172458048220223, −16.507843040410420637113816181240, −15.70001741294724763458325609279, −15.03838087181483131681948426646, −14.63360799203591059166590926204, −13.82287087565780576620229214223, −13.42413463442626345029226299867, −12.67261075985253682274503583716, −12.03848863784209812128188135688, −11.45880357545383112366442584099, −10.22488377023298281657924233060, −9.420505403466788024764480231201, −8.728125597322472821114254245049, −8.07376632477539616020930057392, −7.4806469601798226393352864560, −6.70872863647768169422974490998, −6.2386592745053922546772932648, −5.254874688252290104273148514350, −4.32952452585008995217905967344, −3.82687491248473194336460631072, −2.89414039407597067194561167554, −2.26506070785965697155120292187, −1.29143817600651510348898963872, 0.47921362861088165937395464408, 1.733106012979504105552160530717, 2.39183407396711208702244089229, 3.204123766252898685074531909551, 3.637009837600513393408210596153, 4.76101132909997537482549320572, 4.978175955947312149494046739412, 6.023177272160443906951531678488, 6.90449348583768262927828504927, 7.83287725251942492263354809043, 8.54807334719165694242278833481, 9.328446631936791217430386124700, 10.1140396569130105912304449297, 10.3991302317218146339972136115, 11.2064176879507453013384972618, 12.25678534221616382716786050683, 12.590147111383390745288496981278, 13.46211273519307778711715203839, 14.05171160472963539570008542602, 14.75227911996262151995759234387, 15.03083400107155808405268411421, 15.9845899285492878059043103584, 16.47756012422966400433356302225, 17.58181191814269267186828894429, 18.41668298720442312347934249253

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