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Properties

Label 1-4235-4235.1178-r1-0-0
Degree $1$
Conductor $4235$
Sign $0.825 + 0.564i$
Analytic cond. $455.113$
Root an. cond. $455.113$
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  + (0.971 − 0.235i)2-s + (−0.866 − 0.5i)3-s + (0.888 − 0.458i)4-s + (−0.959 − 0.281i)6-s + (0.755 − 0.654i)8-s + (0.5 + 0.866i)9-s + (−0.998 − 0.0475i)12-s + (−0.540 + 0.841i)13-s + (0.580 − 0.814i)16-s + (0.371 − 0.928i)17-s + (0.690 + 0.723i)18-s + (−0.928 + 0.371i)19-s + (0.814 + 0.580i)23-s + (−0.981 + 0.189i)24-s + (−0.327 + 0.945i)26-s − i·27-s + ⋯
L(s)  = 1  + (0.971 − 0.235i)2-s + (−0.866 − 0.5i)3-s + (0.888 − 0.458i)4-s + (−0.959 − 0.281i)6-s + (0.755 − 0.654i)8-s + (0.5 + 0.866i)9-s + (−0.998 − 0.0475i)12-s + (−0.540 + 0.841i)13-s + (0.580 − 0.814i)16-s + (0.371 − 0.928i)17-s + (0.690 + 0.723i)18-s + (−0.928 + 0.371i)19-s + (0.814 + 0.580i)23-s + (−0.981 + 0.189i)24-s + (−0.327 + 0.945i)26-s − i·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.825 + 0.564i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.825 + 0.564i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.825 + 0.564i$
Analytic conductor: \(455.113\)
Root analytic conductor: \(455.113\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1178, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (1:\ ),\ 0.825 + 0.564i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.241922716 + 0.6938408519i\)
\(L(\frac12)\) \(\approx\) \(2.241922716 + 0.6938408519i\)
\(L(1)\) \(\approx\) \(1.368317806 - 0.3268723975i\)
\(L(1)\) \(\approx\) \(1.368317806 - 0.3268723975i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.971 - 0.235i)T \)
3 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + (-0.540 + 0.841i)T \)
17 \( 1 + (0.371 - 0.928i)T \)
19 \( 1 + (-0.928 + 0.371i)T \)
23 \( 1 + (0.814 + 0.580i)T \)
29 \( 1 + (0.142 - 0.989i)T \)
31 \( 1 + (0.0475 + 0.998i)T \)
37 \( 1 + (-0.458 + 0.888i)T \)
41 \( 1 + (-0.959 - 0.281i)T \)
43 \( 1 + (0.755 - 0.654i)T \)
47 \( 1 + (0.690 - 0.723i)T \)
53 \( 1 + (-0.814 + 0.580i)T \)
59 \( 1 + (-0.235 + 0.971i)T \)
61 \( 1 + (0.723 + 0.690i)T \)
67 \( 1 + (-0.690 - 0.723i)T \)
71 \( 1 + (-0.142 + 0.989i)T \)
73 \( 1 + (0.0950 - 0.995i)T \)
79 \( 1 + (-0.981 - 0.189i)T \)
83 \( 1 + (0.909 + 0.415i)T \)
89 \( 1 + (0.786 - 0.618i)T \)
97 \( 1 + (-0.755 + 0.654i)T \)
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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

−17.77350349268858556410766268139, −17.22458289156500885640960701302, −16.80626211588364808412815612935, −15.99195447006244387150567233609, −15.401141734495730400920069496415, −14.77036049217950104008804412597, −14.361031814704943451544652608239, −12.99816627879141112377176076315, −12.814013711988375684456439877543, −12.17421279102818124379445463529, −11.28199046937727753074941788287, −10.723207678622273505206581927011, −10.25820141539463565577367050681, −9.220959358446289685010989563402, −8.284811871822276757422859820468, −7.483397215391986881933011333361, −6.659568929851048255476552158362, −6.12361953982156873582369667242, −5.34855065868729552535075399381, −4.81607680365214361017442872422, −4.07042289808218409141144738408, −3.344156955081588750524489879768, −2.48645386746979998787497879954, −1.41599587457399299865678992811, −0.30152069259845730819645253451, 0.80498226959103705573000170873, 1.66278061699806524652338732204, 2.36158799989225941093150168041, 3.27534003986275143465137485337, 4.32691749480204916657402924999, 4.832011889384789613896544068834, 5.56107379094084460349056591586, 6.238202361532045757281368373963, 7.08209642901850020463856980524, 7.30028734082826743569707987175, 8.47387418264115184494916433507, 9.56076368756747711843233533043, 10.32241021007133860825392257775, 10.89541285215143009332388966953, 11.81574024627637150422722065463, 11.98288828059796708726929493333, 12.71383113655614678450388929547, 13.63155500708619967069511085282, 13.850775200479992027335764666156, 14.85073166792151710953372330629, 15.50988432480211061821013470388, 16.26060230932109385507182501680, 16.88844006105381166568884039753, 17.37898565321237401818771944264, 18.43659667516328600655606534693

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