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Properties

Label 1-4235-4235.849-r1-0-0
Degree $1$
Conductor $4235$
Sign $-0.138 + 0.990i$
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  + (0.449 + 0.893i)2-s + (−0.669 − 0.743i)3-s + (−0.595 + 0.803i)4-s + (0.362 − 0.931i)6-s + (−0.985 − 0.170i)8-s + (−0.104 + 0.994i)9-s + (0.995 − 0.0950i)12-s + (−0.870 + 0.491i)13-s + (−0.290 − 0.956i)16-s + (−0.179 + 0.983i)17-s + (−0.935 + 0.353i)18-s + (−0.879 + 0.475i)19-s + (0.327 − 0.945i)23-s + (0.532 + 0.846i)24-s + (−0.830 − 0.556i)26-s + (0.809 − 0.587i)27-s + ⋯
L(s)  = 1  + (0.449 + 0.893i)2-s + (−0.669 − 0.743i)3-s + (−0.595 + 0.803i)4-s + (0.362 − 0.931i)6-s + (−0.985 − 0.170i)8-s + (−0.104 + 0.994i)9-s + (0.995 − 0.0950i)12-s + (−0.870 + 0.491i)13-s + (−0.290 − 0.956i)16-s + (−0.179 + 0.983i)17-s + (−0.935 + 0.353i)18-s + (−0.879 + 0.475i)19-s + (0.327 − 0.945i)23-s + (0.532 + 0.846i)24-s + (−0.830 − 0.556i)26-s + (0.809 − 0.587i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.138 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.138 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.138 + 0.990i$
Analytic conductor: \(455.113\)
Root analytic conductor: \(455.113\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (849, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (1:\ ),\ -0.138 + 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8160399613 + 0.9378400983i\)
\(L(\frac12)\) \(\approx\) \(0.8160399613 + 0.9378400983i\)
\(L(1)\) \(\approx\) \(0.7914738436 + 0.3351267177i\)
\(L(1)\) \(\approx\) \(0.7914738436 + 0.3351267177i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.449 + 0.893i)T \)
3 \( 1 + (-0.669 - 0.743i)T \)
13 \( 1 + (-0.870 + 0.491i)T \)
17 \( 1 + (-0.179 + 0.983i)T \)
19 \( 1 + (-0.879 + 0.475i)T \)
23 \( 1 + (0.327 - 0.945i)T \)
29 \( 1 + (0.564 + 0.825i)T \)
31 \( 1 + (0.749 + 0.662i)T \)
37 \( 1 + (-0.953 - 0.299i)T \)
41 \( 1 + (0.254 + 0.967i)T \)
43 \( 1 + (-0.142 - 0.989i)T \)
47 \( 1 + (0.935 + 0.353i)T \)
53 \( 1 + (0.290 - 0.956i)T \)
59 \( 1 + (-0.710 - 0.703i)T \)
61 \( 1 + (-0.548 - 0.836i)T \)
67 \( 1 + (-0.0475 - 0.998i)T \)
71 \( 1 + (0.610 - 0.791i)T \)
73 \( 1 + (0.483 + 0.875i)T \)
79 \( 1 + (0.969 + 0.244i)T \)
83 \( 1 + (0.0855 - 0.996i)T \)
89 \( 1 + (0.235 + 0.971i)T \)
97 \( 1 + (0.466 - 0.884i)T \)
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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

−17.92466664321247446531488498411, −17.40704120000538824707150274260, −16.8649532734503862933299889279, −15.6604270428216279256894270721, −15.3815964898566695733441017616, −14.6958309101990867926342729713, −13.78976511733055671428998562449, −13.252233773973761908226431338381, −12.19773877618000110396755479936, −11.98369416697861935131650496057, −11.15309594982156100229630477006, −10.56459744481900405078732161950, −9.92476992494849610238460603931, −9.34239710015964136781254124610, −8.68264362555040373328306220522, −7.5002371567107124073698209352, −6.57556931090420386772313651971, −5.79246339255967543278581791894, −5.118570771902170570319545589899, −4.54265255915950840885120815864, −3.892623408363929617485616670762, −2.90665121983815476094020621570, −2.38306122398530636606455685818, −1.061125051488589551079255930, −0.345113751392086017603281882359, 0.48549348528748109866348170610, 1.71986751483885417315336074941, 2.52785992103772823397234465018, 3.58574379736957135851406363360, 4.585595115189631785191399141337, 4.98830773564598429217981559889, 5.909192821542876511668574900983, 6.64324720112631671033073189773, 6.869811568469547582827512736894, 7.91609843860453931154245062753, 8.37499737250313092153079527946, 9.172923121270490523865075430327, 10.29803000610386955326091177440, 10.86639083903907834225286874598, 11.93011300954511258215810964120, 12.52184237934165529364533753830, 12.71834020143023251496530467491, 13.80036586355813409438130674236, 14.22768729685329873462154279867, 14.98404252954357512276261815013, 15.701686768217149969884437539883, 16.63526634984566529404897822744, 16.8915539240019033266586198414, 17.55022922980309548759333104330, 18.16479748313911910062909131246

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