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Properties

Label 1-4235-4235.1054-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.532 + 0.846i$
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  + (0.905 + 0.424i)2-s + (0.978 − 0.207i)3-s + (0.640 + 0.768i)4-s + (0.974 + 0.226i)6-s + (0.254 + 0.967i)8-s + (0.913 − 0.406i)9-s + (0.786 + 0.618i)12-s + (−0.696 + 0.717i)13-s + (−0.179 + 0.983i)16-s + (−0.00951 − 0.999i)17-s + (0.999 + 0.0190i)18-s + (0.953 − 0.299i)19-s + (−0.723 + 0.690i)23-s + (0.449 + 0.893i)24-s + (−0.935 + 0.353i)26-s + (0.809 − 0.587i)27-s + ⋯
L(s)  = 1  + (0.905 + 0.424i)2-s + (0.978 − 0.207i)3-s + (0.640 + 0.768i)4-s + (0.974 + 0.226i)6-s + (0.254 + 0.967i)8-s + (0.913 − 0.406i)9-s + (0.786 + 0.618i)12-s + (−0.696 + 0.717i)13-s + (−0.179 + 0.983i)16-s + (−0.00951 − 0.999i)17-s + (0.999 + 0.0190i)18-s + (0.953 − 0.299i)19-s + (−0.723 + 0.690i)23-s + (0.449 + 0.893i)24-s + (−0.935 + 0.353i)26-s + (0.809 − 0.587i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.532 + 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.532 + 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.532 + 0.846i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1054, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.532 + 0.846i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.362477379 + 2.408558890i\)
\(L(\frac12)\) \(\approx\) \(4.362477379 + 2.408558890i\)
\(L(1)\) \(\approx\) \(2.470358725 + 0.7506038651i\)
\(L(1)\) \(\approx\) \(2.470358725 + 0.7506038651i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.905 + 0.424i)T \)
3 \( 1 + (0.978 - 0.207i)T \)
13 \( 1 + (-0.696 + 0.717i)T \)
17 \( 1 + (-0.00951 - 0.999i)T \)
19 \( 1 + (0.953 - 0.299i)T \)
23 \( 1 + (-0.723 + 0.690i)T \)
29 \( 1 + (0.993 + 0.113i)T \)
31 \( 1 + (0.999 - 0.0380i)T \)
37 \( 1 + (0.0665 - 0.997i)T \)
41 \( 1 + (-0.921 + 0.389i)T \)
43 \( 1 + (-0.841 + 0.540i)T \)
47 \( 1 + (0.999 - 0.0190i)T \)
53 \( 1 + (0.179 + 0.983i)T \)
59 \( 1 + (0.123 + 0.992i)T \)
61 \( 1 + (0.820 + 0.572i)T \)
67 \( 1 + (0.327 - 0.945i)T \)
71 \( 1 + (0.198 + 0.980i)T \)
73 \( 1 + (0.851 - 0.524i)T \)
79 \( 1 + (0.988 + 0.151i)T \)
83 \( 1 + (-0.610 + 0.791i)T \)
89 \( 1 + (-0.995 - 0.0950i)T \)
97 \( 1 + (0.998 + 0.0570i)T \)
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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

−18.63907176799854894152074665724, −17.63481552427956297956152890191, −16.73657267260539681091974791767, −15.83891886278219088375608321571, −15.41507320460935634706316319099, −14.765064709928402695940804538886, −14.13693245091786196562276508549, −13.62293129438360878721131197808, −12.8833773519930210586556122867, −12.25597983992749966151351717799, −11.66626617784633237640287408958, −10.440256153906143091341866283862, −10.201832669990827482365301350528, −9.57987694256715869392512108662, −8.395132597690627719413016281289, −7.99413434562216796254667118508, −6.983279821872783243679295068367, −6.35742001035630957077166286881, −5.30464908483481866437402710824, −4.76116558262576575383905970334, −3.8776188779281217446671917991, −3.307689799585084699882326527161, −2.54799409733842752461132108014, −1.89092677102937609396746440094, −0.90187160613408357856479407008, 1.1593696086816389127609807440, 2.22082678836005491262950159840, 2.76137414484409326240971144059, 3.49860587815915293027096657396, 4.35042557240654953748887276205, 4.91419193887072915594897156071, 5.82600374354006415171979771424, 6.84660904751597712891563546054, 7.18077716023150562054221368949, 7.91871410910447017546613467650, 8.6266240635885000713249288492, 9.48171630213058856912163531781, 10.04238348285409795288895184177, 11.23471184149297148821008174692, 12.00170605008953699635923366086, 12.31633472587197490050996179822, 13.39798356534490439580694700658, 13.873464536579095498380031795041, 14.15795657443858199325501781128, 15.06800764102293257096400647379, 15.624108199395571757848049345634, 16.18500412899191512349619060458, 16.94395190080327005786676403246, 17.88635812381641518464234510304, 18.367142009848313209277652939817

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