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Properties

Label 1-4235-4235.873-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.999 + 0.0355i$
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  + (−0.556 + 0.830i)2-s + (−0.406 − 0.913i)3-s + (−0.380 − 0.924i)4-s + (0.985 + 0.170i)6-s + (0.980 + 0.198i)8-s + (−0.669 + 0.743i)9-s + (−0.690 + 0.723i)12-s + (−0.825 + 0.564i)13-s + (−0.710 + 0.703i)16-s + (0.992 + 0.123i)17-s + (−0.244 − 0.969i)18-s + (−0.683 − 0.730i)19-s + (0.458 − 0.888i)23-s + (−0.217 − 0.976i)24-s + (−0.00951 − 0.999i)26-s + (0.951 + 0.309i)27-s + ⋯
L(s)  = 1  + (−0.556 + 0.830i)2-s + (−0.406 − 0.913i)3-s + (−0.380 − 0.924i)4-s + (0.985 + 0.170i)6-s + (0.980 + 0.198i)8-s + (−0.669 + 0.743i)9-s + (−0.690 + 0.723i)12-s + (−0.825 + 0.564i)13-s + (−0.710 + 0.703i)16-s + (0.992 + 0.123i)17-s + (−0.244 − 0.969i)18-s + (−0.683 − 0.730i)19-s + (0.458 − 0.888i)23-s + (−0.217 − 0.976i)24-s + (−0.00951 − 0.999i)26-s + (0.951 + 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0355i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0355i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.999 + 0.0355i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (873, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.999 + 0.0355i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(6.061349114\times10^{-5} + 0.003408471116i\)
\(L(\frac12)\) \(\approx\) \(6.061349114\times10^{-5} + 0.003408471116i\)
\(L(1)\) \(\approx\) \(0.5663948306 + 0.02663440821i\)
\(L(1)\) \(\approx\) \(0.5663948306 + 0.02663440821i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.556 + 0.830i)T \)
3 \( 1 + (-0.406 - 0.913i)T \)
13 \( 1 + (-0.825 + 0.564i)T \)
17 \( 1 + (0.992 + 0.123i)T \)
19 \( 1 + (-0.683 - 0.730i)T \)
23 \( 1 + (0.458 - 0.888i)T \)
29 \( 1 + (-0.0855 + 0.996i)T \)
31 \( 1 + (-0.879 - 0.475i)T \)
37 \( 1 + (-0.647 + 0.761i)T \)
41 \( 1 + (0.466 - 0.884i)T \)
43 \( 1 + (-0.909 + 0.415i)T \)
47 \( 1 + (0.244 - 0.969i)T \)
53 \( 1 + (0.703 - 0.710i)T \)
59 \( 1 + (0.999 + 0.0380i)T \)
61 \( 1 + (0.0665 + 0.997i)T \)
67 \( 1 + (0.371 - 0.928i)T \)
71 \( 1 + (0.516 + 0.856i)T \)
73 \( 1 + (-0.780 + 0.625i)T \)
79 \( 1 + (0.398 + 0.917i)T \)
83 \( 1 + (-0.633 - 0.774i)T \)
89 \( 1 + (-0.327 + 0.945i)T \)
97 \( 1 + (-0.676 - 0.736i)T \)
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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

−18.83817907303342007009803959468, −17.88043405774101992035824020096, −17.41987058937713102421901750722, −16.76558105545282527873500679418, −16.29725349133728435305088160854, −15.40302043472784231846307735625, −14.70515586511706874840473457772, −14.018272993950470194196368329449, −12.989227845251453284830180987986, −12.349860191048430725706421345559, −11.80968964113200102773213959762, −11.044519395698961125127868434650, −10.446137703348066709904247248651, −9.82805178320750322683935848278, −9.38658432901781993139102956359, −8.51225473467815146076914352407, −7.79598056856784004688279850568, −7.05862297167418669960464169867, −5.8561264554569741676586301748, −5.23891223577957853153221278253, −4.41996099401861677019242724364, −3.63913203826965813315272769043, −3.086407821496606189600022955307, −2.148394334665203163768082306002, −1.0938657654261661282246398682, 0.00151234679509252895381259197, 0.99978515902376077089856112818, 1.87956044120117463529235718258, 2.63172001722225271947502203221, 3.975752711158214015221633991504, 5.054712249997230868744600489689, 5.37603361168793842147819235082, 6.35800885605837423196999776023, 7.03235858313137092429047886526, 7.284395716800164010673039801636, 8.36573755117605722130984357020, 8.71470801822255839226320604261, 9.69802066759187986098621400808, 10.428141988861850318951131753571, 11.11151951089465740903235931346, 11.90884820496982079764789656499, 12.68658367585147656770532621643, 13.28179644892216248773733278607, 14.15589119393012467375479454295, 14.62873157728993856122387824175, 15.27273452276395251511123756338, 16.45802217156773255558139161116, 16.660736082623130851312663553139, 17.293088391948755639424158773753, 18.00185645594396353028004509544

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