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Properties

Label 1-4235-4235.1234-r1-0-0
Degree $1$
Conductor $4235$
Sign $0.332 + 0.943i$
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  + (0.999 − 0.0380i)2-s + (−0.669 − 0.743i)3-s + (0.997 − 0.0760i)4-s + (−0.696 − 0.717i)6-s + (0.993 − 0.113i)8-s + (−0.104 + 0.994i)9-s + (−0.723 − 0.690i)12-s + (0.941 + 0.336i)13-s + (0.988 − 0.151i)16-s + (−0.683 + 0.730i)17-s + (−0.0665 + 0.997i)18-s + (−0.483 + 0.875i)19-s + (0.888 + 0.458i)23-s + (−0.749 − 0.662i)24-s + (0.953 + 0.299i)26-s + (0.809 − 0.587i)27-s + ⋯
L(s)  = 1  + (0.999 − 0.0380i)2-s + (−0.669 − 0.743i)3-s + (0.997 − 0.0760i)4-s + (−0.696 − 0.717i)6-s + (0.993 − 0.113i)8-s + (−0.104 + 0.994i)9-s + (−0.723 − 0.690i)12-s + (0.941 + 0.336i)13-s + (0.988 − 0.151i)16-s + (−0.683 + 0.730i)17-s + (−0.0665 + 0.997i)18-s + (−0.483 + 0.875i)19-s + (0.888 + 0.458i)23-s + (−0.749 − 0.662i)24-s + (0.953 + 0.299i)26-s + (0.809 − 0.587i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.332 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.332 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.332 + 0.943i$
Analytic conductor: \(455.113\)
Root analytic conductor: \(455.113\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1234, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (1:\ ),\ 0.332 + 0.943i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.706002472 + 1.915009420i\)
\(L(\frac12)\) \(\approx\) \(2.706002472 + 1.915009420i\)
\(L(1)\) \(\approx\) \(1.671706843 - 0.07024742319i\)
\(L(1)\) \(\approx\) \(1.671706843 - 0.07024742319i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.999 - 0.0380i)T \)
3 \( 1 + (-0.669 - 0.743i)T \)
13 \( 1 + (0.941 + 0.336i)T \)
17 \( 1 + (-0.683 + 0.730i)T \)
19 \( 1 + (-0.483 + 0.875i)T \)
23 \( 1 + (0.888 + 0.458i)T \)
29 \( 1 + (0.921 + 0.389i)T \)
31 \( 1 + (-0.991 + 0.132i)T \)
37 \( 1 + (0.851 - 0.524i)T \)
41 \( 1 + (0.985 + 0.170i)T \)
43 \( 1 + (0.415 + 0.909i)T \)
47 \( 1 + (0.0665 + 0.997i)T \)
53 \( 1 + (-0.988 - 0.151i)T \)
59 \( 1 + (0.345 - 0.938i)T \)
61 \( 1 + (0.532 - 0.846i)T \)
67 \( 1 + (-0.928 - 0.371i)T \)
71 \( 1 + (0.0855 - 0.996i)T \)
73 \( 1 + (-0.935 + 0.353i)T \)
79 \( 1 + (-0.861 - 0.508i)T \)
83 \( 1 + (-0.998 + 0.0570i)T \)
89 \( 1 + (-0.327 + 0.945i)T \)
97 \( 1 + (-0.198 + 0.980i)T \)
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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

−17.87066873970864589913880906254, −17.2527982099058402097694860445, −16.47641681694183608256693604792, −15.92868864840669612396109522495, −15.38701526724360133220372297051, −14.83161813603116277887328006247, −14.0209612151162903066328888, −13.191260210816385126519152780, −12.77486826987088257210082417837, −11.78466097865479705479223899087, −11.28868979960357478698794661067, −10.763920225477791436883245275037, −10.115801159401195363182975232627, −9.07656453449497717733097035246, −8.48767636671706777911519751562, −7.21675762022637472877283954477, −6.738695797198767760415341704935, −5.896026859203436961971316125570, −5.407208582862784446244318835662, −4.407185030362226883901506816538, −4.23313667849346733799603096802, −3.09172761375506808787976171700, −2.56504432251859804750583774035, −1.24540410951202267840520448470, −0.35942200157766980570432918693, 1.07629387102467374221150212264, 1.62596107289679057896342815371, 2.47165800725332245940289202807, 3.42530677647302244326824766068, 4.271675567252136770054077671297, 4.916819547109108559015034881932, 5.94202443414308711125690177388, 6.16943556851295763169016859814, 6.92934492091689387583193588877, 7.71742909208333404801361647631, 8.37575771760388981976677002940, 9.424780432433395641185963809016, 10.64022101241730317326320704765, 10.955494002666012610835982752, 11.51811344334988755935097055205, 12.45977307168302651406719307584, 12.87030586958680413487907866993, 13.34949483719126773644780625958, 14.29107745208026439323372415673, 14.65278114321625890699414753784, 15.8279939817740507580989703073, 16.12155844371043653481966990818, 16.9412533434831527161843408924, 17.58201756230968786972053209179, 18.34101337850382026670526381285

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