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Properties

Label 1-4235-4235.3048-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.974 + 0.224i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.189 − 0.981i)2-s + (−0.866 + 0.5i)3-s + (−0.928 + 0.371i)4-s + (0.654 + 0.755i)6-s + (0.540 + 0.841i)8-s + (0.5 − 0.866i)9-s + (0.618 − 0.786i)12-s + (0.989 + 0.142i)13-s + (0.723 − 0.690i)16-s + (0.814 + 0.580i)17-s + (−0.945 − 0.327i)18-s + (0.580 + 0.814i)19-s + (0.690 + 0.723i)23-s + (−0.888 − 0.458i)24-s + (−0.0475 − 0.998i)26-s + i·27-s + ⋯
L(s)  = 1  + (−0.189 − 0.981i)2-s + (−0.866 + 0.5i)3-s + (−0.928 + 0.371i)4-s + (0.654 + 0.755i)6-s + (0.540 + 0.841i)8-s + (0.5 − 0.866i)9-s + (0.618 − 0.786i)12-s + (0.989 + 0.142i)13-s + (0.723 − 0.690i)16-s + (0.814 + 0.580i)17-s + (−0.945 − 0.327i)18-s + (0.580 + 0.814i)19-s + (0.690 + 0.723i)23-s + (−0.888 − 0.458i)24-s + (−0.0475 − 0.998i)26-s + i·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.974 + 0.224i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.974 + 0.224i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.974 + 0.224i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (3048, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.974 + 0.224i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.270399190 + 0.1445506063i\)
\(L(\frac12)\) \(\approx\) \(1.270399190 + 0.1445506063i\)
\(L(1)\) \(\approx\) \(0.8169231825 - 0.1218947272i\)
\(L(1)\) \(\approx\) \(0.8169231825 - 0.1218947272i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.189 - 0.981i)T \)
3 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + (0.989 + 0.142i)T \)
17 \( 1 + (0.814 + 0.580i)T \)
19 \( 1 + (0.580 + 0.814i)T \)
23 \( 1 + (0.690 + 0.723i)T \)
29 \( 1 + (-0.415 + 0.909i)T \)
31 \( 1 + (0.786 - 0.618i)T \)
37 \( 1 + (0.371 - 0.928i)T \)
41 \( 1 + (0.654 + 0.755i)T \)
43 \( 1 + (0.540 + 0.841i)T \)
47 \( 1 + (0.945 - 0.327i)T \)
53 \( 1 + (-0.690 + 0.723i)T \)
59 \( 1 + (0.981 + 0.189i)T \)
61 \( 1 + (0.327 + 0.945i)T \)
67 \( 1 + (0.945 + 0.327i)T \)
71 \( 1 + (0.415 - 0.909i)T \)
73 \( 1 + (0.971 - 0.235i)T \)
79 \( 1 + (0.888 - 0.458i)T \)
83 \( 1 + (0.281 + 0.959i)T \)
89 \( 1 + (-0.995 - 0.0950i)T \)
97 \( 1 + (0.540 + 0.841i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.2614562592642048156704648067, −17.48280662699010839610495660679, −17.13289782386688034436501483855, −16.292042886369223902162091904850, −15.812294815015921411481904149167, −15.24451410198374934219838602738, −14.09838785318977664242135636855, −13.76274761360210963611179228059, −12.96806104792602137711330256097, −12.37797326227006686912876424846, −11.43958770750526420172891174558, −10.84407891874489031178934828333, −10.02142122854315898762738533048, −9.33660587778708955120006829215, −8.408657308257083452183654595, −7.84772503160600051884130826965, −7.0059189185509166890011418050, −6.58103300193977322596977855399, −5.73978181562771573993841183712, −5.2178954193917833356257101030, −4.502002708617501690120367790, −3.56118654169737972294628859017, −2.39795043768736125857776435143, −1.08199506422978584061586725773, −0.66526149724291082103714620976, 0.99418306475521488283675850495, 1.34091578112031553124093581675, 2.6278654100203375328241695227, 3.69679902089491930815755258039, 3.86233440022289790411977875706, 4.94662056731778527766849407666, 5.61367970847746776992214159243, 6.22532840930279091135296591133, 7.40599943707520122222866460740, 8.09706596533063426266840073707, 9.07584824476710433273204454098, 9.57340094601259814588177804501, 10.28641592359893876170849212158, 10.97750030359823992700565251448, 11.35921389299327451098706141506, 12.189685188869479921249083300057, 12.6881727886848818438940224558, 13.43300525276127328666553879514, 14.27692417963219516253009039151, 14.967957712009356107713584057489, 15.91095428744352522010601741460, 16.55838804005084686850485607667, 17.05936444060609477227629468121, 17.90811767679191594664402485036, 18.31805677252915880660109011548

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