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Properties

Label 1-4235-4235.527-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.977 - 0.212i$
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  + (0.618 + 0.786i)2-s + (0.866 + 0.5i)3-s + (−0.235 + 0.971i)4-s + (0.142 + 0.989i)6-s + (−0.909 + 0.415i)8-s + (0.5 + 0.866i)9-s + (−0.690 + 0.723i)12-s + (−0.281 − 0.959i)13-s + (−0.888 − 0.458i)16-s + (−0.189 − 0.981i)17-s + (−0.371 + 0.928i)18-s + (0.981 + 0.189i)19-s + (−0.458 + 0.888i)23-s + (−0.995 − 0.0950i)24-s + (0.580 − 0.814i)26-s + i·27-s + ⋯
L(s)  = 1  + (0.618 + 0.786i)2-s + (0.866 + 0.5i)3-s + (−0.235 + 0.971i)4-s + (0.142 + 0.989i)6-s + (−0.909 + 0.415i)8-s + (0.5 + 0.866i)9-s + (−0.690 + 0.723i)12-s + (−0.281 − 0.959i)13-s + (−0.888 − 0.458i)16-s + (−0.189 − 0.981i)17-s + (−0.371 + 0.928i)18-s + (0.981 + 0.189i)19-s + (−0.458 + 0.888i)23-s + (−0.995 − 0.0950i)24-s + (0.580 − 0.814i)26-s + i·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.212i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.212i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.977 - 0.212i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (527, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.977 - 0.212i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2569804399 + 2.386735395i\)
\(L(\frac12)\) \(\approx\) \(-0.2569804399 + 2.386735395i\)
\(L(1)\) \(\approx\) \(1.167575375 + 1.179192442i\)
\(L(1)\) \(\approx\) \(1.167575375 + 1.179192442i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.618 + 0.786i)T \)
3 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (-0.281 - 0.959i)T \)
17 \( 1 + (-0.189 - 0.981i)T \)
19 \( 1 + (0.981 + 0.189i)T \)
23 \( 1 + (-0.458 + 0.888i)T \)
29 \( 1 + (-0.654 + 0.755i)T \)
31 \( 1 + (0.723 - 0.690i)T \)
37 \( 1 + (-0.971 + 0.235i)T \)
41 \( 1 + (0.142 + 0.989i)T \)
43 \( 1 + (-0.909 + 0.415i)T \)
47 \( 1 + (0.371 + 0.928i)T \)
53 \( 1 + (0.458 + 0.888i)T \)
59 \( 1 + (0.786 + 0.618i)T \)
61 \( 1 + (-0.928 + 0.371i)T \)
67 \( 1 + (-0.371 + 0.928i)T \)
71 \( 1 + (-0.654 + 0.755i)T \)
73 \( 1 + (-0.998 + 0.0475i)T \)
79 \( 1 + (-0.995 + 0.0950i)T \)
83 \( 1 + (-0.540 + 0.841i)T \)
89 \( 1 + (0.327 - 0.945i)T \)
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

−18.238910410596525035157250173823, −17.62616559422918277684882923164, −16.624637663497522904868552099478, −15.63218600623530835976934760214, −15.13297502624292298614365831664, −14.33061080147466205968305744075, −13.91240244376828212062792724172, −13.3295404247393392831225773326, −12.54169171445199120394693262010, −12.01186739370866831088201098991, −11.42000893478029267151318300526, −10.31178456843864471192111292317, −9.93488535088854464139222744759, −8.92738289998219836175314758261, −8.620275211375107115347051759776, −7.49637990822332208647524507762, −6.755491276180645954842378279470, −6.121787710645509389655667801592, −5.14455813484340172084284011531, −4.26777684535949707193127570455, −3.68104415250700816638866887123, −2.9368829435544421191530221446, −1.98530130832296076603320955344, −1.68873850159057955989287483033, −0.42673469152128021822657739765, 1.35391607373775026310823239853, 2.70070656198535882401783675340, 3.02366102637888845814074993396, 3.86133012912771087834252145449, 4.64954609971852474066229126743, 5.3111744857009647050907593802, 5.93268869429422024999723926678, 7.19545738995551555347469992085, 7.48753919322074623649799413582, 8.22223243193169111201650246509, 8.96773398855031990794273192269, 9.6458577620851362328743768432, 10.280897180156183319614129735, 11.41355781803293267103697843881, 11.98040733522161328846536125162, 13.02127686033708381876301011772, 13.40904509037618505340067086677, 14.10829599158175609655902521899, 14.6701744817461191839669081999, 15.40041840365271788660079231447, 15.8239709375288666977391324354, 16.42777670197030413456967926917, 17.22763543414358684535060366751, 18.01444227447641893358356361731, 18.55821906499114843982844602321

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