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Properties

Label 1-4235-4235.1013-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.718 + 0.695i$
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  + (−0.371 − 0.928i)2-s + (0.866 + 0.5i)3-s + (−0.723 + 0.690i)4-s + (0.142 − 0.989i)6-s + (0.909 + 0.415i)8-s + (0.5 + 0.866i)9-s + (−0.971 + 0.235i)12-s + (−0.281 + 0.959i)13-s + (0.0475 − 0.998i)16-s + (0.945 − 0.327i)17-s + (0.618 − 0.786i)18-s + (−0.327 + 0.945i)19-s + (0.998 + 0.0475i)23-s + (0.580 + 0.814i)24-s + (0.995 − 0.0950i)26-s + i·27-s + ⋯
L(s)  = 1  + (−0.371 − 0.928i)2-s + (0.866 + 0.5i)3-s + (−0.723 + 0.690i)4-s + (0.142 − 0.989i)6-s + (0.909 + 0.415i)8-s + (0.5 + 0.866i)9-s + (−0.971 + 0.235i)12-s + (−0.281 + 0.959i)13-s + (0.0475 − 0.998i)16-s + (0.945 − 0.327i)17-s + (0.618 − 0.786i)18-s + (−0.327 + 0.945i)19-s + (0.998 + 0.0475i)23-s + (0.580 + 0.814i)24-s + (0.995 − 0.0950i)26-s + i·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.718 + 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.718 + 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.718 + 0.695i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1013, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.718 + 0.695i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.781472604 + 0.7211746516i\)
\(L(\frac12)\) \(\approx\) \(1.781472604 + 0.7211746516i\)
\(L(1)\) \(\approx\) \(1.183466963 + 0.02139792740i\)
\(L(1)\) \(\approx\) \(1.183466963 + 0.02139792740i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.371 - 0.928i)T \)
3 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (-0.281 + 0.959i)T \)
17 \( 1 + (0.945 - 0.327i)T \)
19 \( 1 + (-0.327 + 0.945i)T \)
23 \( 1 + (0.998 + 0.0475i)T \)
29 \( 1 + (0.654 + 0.755i)T \)
31 \( 1 + (-0.235 + 0.971i)T \)
37 \( 1 + (0.690 - 0.723i)T \)
41 \( 1 + (0.142 - 0.989i)T \)
43 \( 1 + (0.909 + 0.415i)T \)
47 \( 1 + (-0.618 - 0.786i)T \)
53 \( 1 + (-0.998 + 0.0475i)T \)
59 \( 1 + (0.928 + 0.371i)T \)
61 \( 1 + (0.786 - 0.618i)T \)
67 \( 1 + (-0.618 + 0.786i)T \)
71 \( 1 + (-0.654 - 0.755i)T \)
73 \( 1 + (0.458 + 0.888i)T \)
79 \( 1 + (-0.580 + 0.814i)T \)
83 \( 1 + (-0.540 - 0.841i)T \)
89 \( 1 + (0.981 + 0.189i)T \)
97 \( 1 + (0.909 + 0.415i)T \)
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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

−18.236951588293594305603105702585, −17.53785873450734441180286555912, −17.13652858503767525474099167657, −16.15694389914196945916486370445, −15.47167325968395215617809293368, −14.794649262582428265959107007728, −14.58410518746072839697373645785, −13.502365602974612780648441320959, −13.13177018742663395242311361098, −12.490670146295032190189745986976, −11.40748197304022958986058941486, −10.46765416775080715190135089071, −9.73268760706897236701929025895, −9.240679283930377406084524945849, −8.347857517998592483409756066069, −7.90128135179028762369341432559, −7.32380263669036502305098927575, −6.50141250352274216114310688229, −5.895161401300800021709035669412, −4.94046820092051597907982484647, −4.20869850134782796918225518366, −3.1806187483684476041741068563, −2.48397125106006174730331115377, −1.30705248614011106446522830637, −0.61384823538631081471251915085, 1.12018858952136121529102127230, 1.86616664010506746958971760256, 2.68141352612167275106782945328, 3.37024186483627395111728848899, 4.029661278275962880890171772725, 4.76863349552728768373171715425, 5.49617318144007001610665807139, 6.920868244858600509198386547422, 7.523386811828232338114747691167, 8.34827441968869333607777330373, 8.92185554750828536535741890379, 9.52578534917889530338320304739, 10.15933454110948788893747491413, 10.76874791307036658962577703692, 11.5209468600090444788392885623, 12.38960346391403237573329306212, 12.84699153998630142939504650559, 13.7879084631023308446165087982, 14.389176090730376137104283950770, 14.69944726938966847881890560844, 16.06602749131421600775765265741, 16.334018625223267912862087129426, 17.12463991614464529994422270864, 17.94813991443422307062982110925, 18.87138765031620424469269248948

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