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Properties

Label 1-4235-4235.558-r1-0-0
Degree $1$
Conductor $4235$
Sign $0.822 + 0.569i$
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  + (0.353 − 0.935i)2-s + (−0.994 + 0.104i)3-s + (−0.749 − 0.662i)4-s + (−0.254 + 0.967i)6-s + (−0.884 + 0.466i)8-s + (0.978 − 0.207i)9-s + (0.814 + 0.580i)12-s + (0.113 + 0.993i)13-s + (0.123 + 0.992i)16-s + (−0.0760 − 0.997i)17-s + (0.151 − 0.988i)18-s + (0.761 + 0.647i)19-s + (−0.189 + 0.981i)23-s + (0.830 − 0.556i)24-s + (0.969 + 0.244i)26-s + (−0.951 + 0.309i)27-s + ⋯
L(s)  = 1  + (0.353 − 0.935i)2-s + (−0.994 + 0.104i)3-s + (−0.749 − 0.662i)4-s + (−0.254 + 0.967i)6-s + (−0.884 + 0.466i)8-s + (0.978 − 0.207i)9-s + (0.814 + 0.580i)12-s + (0.113 + 0.993i)13-s + (0.123 + 0.992i)16-s + (−0.0760 − 0.997i)17-s + (0.151 − 0.988i)18-s + (0.761 + 0.647i)19-s + (−0.189 + 0.981i)23-s + (0.830 − 0.556i)24-s + (0.969 + 0.244i)26-s + (−0.951 + 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.822 + 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.822 + 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.822 + 0.569i$
Analytic conductor: \(455.113\)
Root analytic conductor: \(455.113\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (558, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (1:\ ),\ 0.822 + 0.569i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.230306265 + 0.3844490664i\)
\(L(\frac12)\) \(\approx\) \(1.230306265 + 0.3844490664i\)
\(L(1)\) \(\approx\) \(0.8121681768 - 0.2845921440i\)
\(L(1)\) \(\approx\) \(0.8121681768 - 0.2845921440i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.353 - 0.935i)T \)
3 \( 1 + (-0.994 + 0.104i)T \)
13 \( 1 + (0.113 + 0.993i)T \)
17 \( 1 + (-0.0760 - 0.997i)T \)
19 \( 1 + (0.761 + 0.647i)T \)
23 \( 1 + (-0.189 + 0.981i)T \)
29 \( 1 + (0.610 + 0.791i)T \)
31 \( 1 + (-0.953 + 0.299i)T \)
37 \( 1 + (0.508 - 0.861i)T \)
41 \( 1 + (-0.998 + 0.0570i)T \)
43 \( 1 + (0.989 + 0.142i)T \)
47 \( 1 + (0.151 + 0.988i)T \)
53 \( 1 + (0.992 + 0.123i)T \)
59 \( 1 + (0.548 - 0.836i)T \)
61 \( 1 + (0.161 - 0.986i)T \)
67 \( 1 + (0.458 + 0.888i)T \)
71 \( 1 + (-0.0285 - 0.999i)T \)
73 \( 1 + (-0.956 - 0.290i)T \)
79 \( 1 + (0.345 + 0.938i)T \)
83 \( 1 + (0.856 + 0.516i)T \)
89 \( 1 + (0.723 + 0.690i)T \)
97 \( 1 + (-0.441 + 0.897i)T \)
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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

−17.9873912464718171248435268329, −17.35273349468061598888565050420, −16.76474362220113859577106730534, −16.19230742998674090263537775741, −15.37046447162674054944611137788, −15.08036626313892279597947811524, −14.083901135353557321451341972, −13.16005074746873296715037663910, −12.97091019013712854962650600509, −12.04018629016654637186969799316, −11.54043676459708878106559445022, −10.42814615873578169675598254446, −10.076098807125974818771838438213, −8.9617912570581871522262474206, −8.26182031952040974875783149262, −7.500280444334768120134624659512, −6.88481156877184255658221788401, −6.062766442862260544119362361220, −5.658426417743418247377190712997, −4.8644260739641758380532556798, −4.20167212493243231960800477728, −3.37901523816210854751552367347, −2.30911848119497352906861814931, −0.949662870955065607355614742801, −0.29317514336687067030423440914, 0.78047139364150005607795091994, 1.46648415386297212428588543858, 2.30550960735963793712349177734, 3.44738163385466458233087444468, 3.990225636048108732951593161661, 4.92290071816840683731355422395, 5.34792708678885026686351175772, 6.15252987296717511448523599112, 6.953847919882224501509552284374, 7.74733778766168278085653195040, 9.045338287344667392881488262841, 9.44495328554989848893313425487, 10.16494842445380419855913962697, 10.95129940381829969085991186153, 11.460834142175569770123056589188, 12.021733613624177983373926743159, 12.58444445796821689907697072608, 13.41521838098156984804663426273, 14.072214566310839025424181684993, 14.66007367302890494910881883108, 15.790956875931872769195733087933, 16.147464818951722502677763518895, 16.99405181262375911479303216498, 17.900235272652709784276480740, 18.19467628411749364877635644752

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