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Properties

Label 1-4235-4235.1439-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.993 - 0.112i$
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  + (−0.00951 + 0.999i)2-s + (0.978 − 0.207i)3-s + (−0.999 − 0.0190i)4-s + (0.198 + 0.980i)6-s + (0.0285 − 0.999i)8-s + (0.913 − 0.406i)9-s + (−0.981 + 0.189i)12-s + (−0.0855 − 0.996i)13-s + (0.999 + 0.0380i)16-s + (−0.548 − 0.836i)17-s + (0.398 + 0.917i)18-s + (0.964 + 0.263i)19-s + (0.786 + 0.618i)23-s + (−0.179 − 0.983i)24-s + (0.997 − 0.0760i)26-s + (0.809 − 0.587i)27-s + ⋯
L(s)  = 1  + (−0.00951 + 0.999i)2-s + (0.978 − 0.207i)3-s + (−0.999 − 0.0190i)4-s + (0.198 + 0.980i)6-s + (0.0285 − 0.999i)8-s + (0.913 − 0.406i)9-s + (−0.981 + 0.189i)12-s + (−0.0855 − 0.996i)13-s + (0.999 + 0.0380i)16-s + (−0.548 − 0.836i)17-s + (0.398 + 0.917i)18-s + (0.964 + 0.263i)19-s + (0.786 + 0.618i)23-s + (−0.179 − 0.983i)24-s + (0.997 − 0.0760i)26-s + (0.809 − 0.587i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.993 - 0.112i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.993 - 0.112i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.236011942 - 0.1261395289i\)
\(L(\frac12)\) \(\approx\) \(2.236011942 - 0.1261395289i\)
\(L(1)\) \(\approx\) \(1.336232576 + 0.3136269893i\)
\(L(1)\) \(\approx\) \(1.336232576 + 0.3136269893i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.00951 + 0.999i)T \)
3 \( 1 + (0.978 - 0.207i)T \)
13 \( 1 + (-0.0855 - 0.996i)T \)
17 \( 1 + (-0.548 - 0.836i)T \)
19 \( 1 + (0.964 + 0.263i)T \)
23 \( 1 + (0.786 + 0.618i)T \)
29 \( 1 + (0.774 + 0.633i)T \)
31 \( 1 + (-0.683 - 0.730i)T \)
37 \( 1 + (-0.797 + 0.603i)T \)
41 \( 1 + (-0.736 - 0.676i)T \)
43 \( 1 + (0.959 - 0.281i)T \)
47 \( 1 + (0.398 - 0.917i)T \)
53 \( 1 + (-0.999 + 0.0380i)T \)
59 \( 1 + (0.953 + 0.299i)T \)
61 \( 1 + (0.861 - 0.508i)T \)
67 \( 1 + (0.995 - 0.0950i)T \)
71 \( 1 + (-0.362 + 0.931i)T \)
73 \( 1 + (-0.640 - 0.768i)T \)
79 \( 1 + (-0.991 + 0.132i)T \)
83 \( 1 + (-0.696 - 0.717i)T \)
89 \( 1 + (-0.888 + 0.458i)T \)
97 \( 1 + (-0.941 - 0.336i)T \)
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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

−18.68055144870122202742013914113, −17.87381572473953486643933590886, −17.23335759453946571007103981945, −16.278806880937655718794859145571, −15.62960588594947827439116477529, −14.69340580222349757750569181199, −14.21385127945583594238890959114, −13.66595585624766858875576125539, −12.869372410412978307545534237152, −12.407316817008208382766727088461, −11.41610081002434397896692096498, −10.86844814959046861130132420608, −10.04869547517442145496149725730, −9.48184648062041365430994733654, −8.78312125245832785853426273849, −8.37389022452574311084593223757, −7.38841814344435747213851193958, −6.64494573897222812792109064623, −5.44161332490445563212296425512, −4.5743654563342185162066654237, −4.10466094857755802107849809032, −3.25089789784165160954545570713, −2.59810861966084955254349054428, −1.83090166076137397898368418622, −1.09192574931829267381137936243, 0.597643507558800173988423520609, 1.5425906429274075933800727629, 2.783451145714676484557702528520, 3.37066573516182915195618945403, 4.1802491755709949655428060451, 5.15346850519089686114887651126, 5.60910928748079543090694518312, 6.91365366700564469511979674260, 7.101586705390691448852201291045, 7.934844019543858639418013256431, 8.5484718302351489844859235810, 9.209220987767337528768465930587, 9.82042903761857948259426111104, 10.52323993221396355693107457477, 11.68917578790438586779123306988, 12.574288323684632664195029580, 13.167441300801673461002660612500, 13.76049563586475089555240453633, 14.32141326975514675477050897188, 15.01179475847153164279944300502, 15.748823883295657904982210467299, 15.92360160124140983898859870349, 17.08618135832802139817880502045, 17.668388301736384057657746150909, 18.34415216189147825918395158703

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