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Properties

Label 1-4235-4235.3659-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.885 - 0.465i$
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  + (−0.432 + 0.901i)2-s + (0.913 − 0.406i)3-s + (−0.625 − 0.780i)4-s + (−0.0285 + 0.999i)6-s + (0.974 − 0.226i)8-s + (0.669 − 0.743i)9-s + (−0.888 − 0.458i)12-s + (−0.774 − 0.633i)13-s + (−0.217 + 0.976i)16-s + (0.830 − 0.556i)17-s + (0.380 + 0.924i)18-s + (0.999 − 0.0380i)19-s + (0.995 − 0.0950i)23-s + (0.797 − 0.603i)24-s + (0.905 − 0.424i)26-s + (0.309 − 0.951i)27-s + ⋯
L(s)  = 1  + (−0.432 + 0.901i)2-s + (0.913 − 0.406i)3-s + (−0.625 − 0.780i)4-s + (−0.0285 + 0.999i)6-s + (0.974 − 0.226i)8-s + (0.669 − 0.743i)9-s + (−0.888 − 0.458i)12-s + (−0.774 − 0.633i)13-s + (−0.217 + 0.976i)16-s + (0.830 − 0.556i)17-s + (0.380 + 0.924i)18-s + (0.999 − 0.0380i)19-s + (0.995 − 0.0950i)23-s + (0.797 − 0.603i)24-s + (0.905 − 0.424i)26-s + (0.309 − 0.951i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.885 - 0.465i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.885 - 0.465i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.885 - 0.465i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (3659, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.885 - 0.465i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.916537195 - 0.4732586147i\)
\(L(\frac12)\) \(\approx\) \(1.916537195 - 0.4732586147i\)
\(L(1)\) \(\approx\) \(1.190787622 + 0.07964497516i\)
\(L(1)\) \(\approx\) \(1.190787622 + 0.07964497516i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.432 + 0.901i)T \)
3 \( 1 + (0.913 - 0.406i)T \)
13 \( 1 + (-0.774 - 0.633i)T \)
17 \( 1 + (0.830 - 0.556i)T \)
19 \( 1 + (0.999 - 0.0380i)T \)
23 \( 1 + (0.995 - 0.0950i)T \)
29 \( 1 + (-0.696 - 0.717i)T \)
31 \( 1 + (0.710 + 0.703i)T \)
37 \( 1 + (-0.548 - 0.836i)T \)
41 \( 1 + (0.941 + 0.336i)T \)
43 \( 1 + (-0.654 + 0.755i)T \)
47 \( 1 + (0.380 - 0.924i)T \)
53 \( 1 + (0.217 + 0.976i)T \)
59 \( 1 + (0.179 - 0.983i)T \)
61 \( 1 + (0.997 + 0.0760i)T \)
67 \( 1 + (-0.235 + 0.971i)T \)
71 \( 1 + (-0.985 - 0.170i)T \)
73 \( 1 + (0.948 + 0.318i)T \)
79 \( 1 + (0.999 + 0.0190i)T \)
83 \( 1 + (-0.993 + 0.113i)T \)
89 \( 1 + (-0.928 + 0.371i)T \)
97 \( 1 + (-0.921 - 0.389i)T \)
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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

−18.77035987964714669389542605504, −17.92480285484407288616877744611, −16.96828131609595986179149810728, −16.63956053724216530627079912852, −15.740237795665029628306220085117, −14.883447295108594269448813766, −14.286473040380059655654798844999, −13.63817528979541407903883195658, −12.96523774611308446289819125422, −12.23741452827888734403084592475, −11.55907158596221284584304654847, −10.737588914447139226998260895106, −10.05089943565387587649861444901, −9.513789416985033137044672534860, −8.96630456549831816564704149738, −8.18901770729440116764761433220, −7.52446346121332593377701287094, −6.937970592665182957613636667136, −5.44334272725093615566444213358, −4.786023674670219533881464904547, −3.9619358985630474651644183375, −3.303894873399452769328135434488, −2.654744198095943562234573742159, −1.82839097137783818215707369761, −1.05710056360937945831872896583, 0.65397568574708767149269395598, 1.37226307533701501377536186904, 2.50768342143389368636031366115, 3.22268524360020720717647672845, 4.186432490240416745288019303584, 5.132357081361565518022857226163, 5.6774885019115160449804894956, 6.783050742183323425214416498179, 7.27929312976246730296995601784, 7.834388619213385857393691243393, 8.472965356858580030359323517819, 9.338697616844749015999478152700, 9.72675926501817122905321667785, 10.430451519408482521640379173711, 11.51256514385803573405287930234, 12.425799214288161528577826615213, 13.06568493265185405881968920480, 13.82157387290087023588729131061, 14.32819884947797656489954665755, 14.96378281381100541140647092133, 15.52365174862202049459985028486, 16.23193226468492241202578601871, 16.98832770643775105179624487521, 17.758710811550065060461296793508, 18.21527197899822317292471784019

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