Properties

Label 1-4235-4235.1647-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.755 + 0.655i$
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  + (−0.353 + 0.935i)2-s + (−0.994 + 0.104i)3-s + (−0.749 − 0.662i)4-s + (0.254 − 0.967i)6-s + (0.884 − 0.466i)8-s + (0.978 − 0.207i)9-s + (0.814 + 0.580i)12-s + (0.113 + 0.993i)13-s + (0.123 + 0.992i)16-s + (−0.0760 − 0.997i)17-s + (−0.151 + 0.988i)18-s + (−0.761 − 0.647i)19-s + (0.189 − 0.981i)23-s + (−0.830 + 0.556i)24-s + (−0.969 − 0.244i)26-s + (−0.951 + 0.309i)27-s + ⋯
L(s)  = 1  + (−0.353 + 0.935i)2-s + (−0.994 + 0.104i)3-s + (−0.749 − 0.662i)4-s + (0.254 − 0.967i)6-s + (0.884 − 0.466i)8-s + (0.978 − 0.207i)9-s + (0.814 + 0.580i)12-s + (0.113 + 0.993i)13-s + (0.123 + 0.992i)16-s + (−0.0760 − 0.997i)17-s + (−0.151 + 0.988i)18-s + (−0.761 − 0.647i)19-s + (0.189 − 0.981i)23-s + (−0.830 + 0.556i)24-s + (−0.969 − 0.244i)26-s + (−0.951 + 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.755 + 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.755 + 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.755 + 0.655i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1647, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.755 + 0.655i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2004255891 + 0.5365557950i\)
\(L(\frac12)\) \(\approx\) \(0.2004255891 + 0.5365557950i\)
\(L(1)\) \(\approx\) \(0.5238120140 + 0.2579624745i\)
\(L(1)\) \(\approx\) \(0.5238120140 + 0.2579624745i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.353 + 0.935i)T \)
3 \( 1 + (-0.994 + 0.104i)T \)
13 \( 1 + (0.113 + 0.993i)T \)
17 \( 1 + (-0.0760 - 0.997i)T \)
19 \( 1 + (-0.761 - 0.647i)T \)
23 \( 1 + (0.189 - 0.981i)T \)
29 \( 1 + (0.610 + 0.791i)T \)
31 \( 1 + (0.953 - 0.299i)T \)
37 \( 1 + (-0.508 + 0.861i)T \)
41 \( 1 + (0.998 - 0.0570i)T \)
43 \( 1 + (-0.989 - 0.142i)T \)
47 \( 1 + (0.151 + 0.988i)T \)
53 \( 1 + (-0.992 - 0.123i)T \)
59 \( 1 + (-0.548 + 0.836i)T \)
61 \( 1 + (-0.161 + 0.986i)T \)
67 \( 1 + (-0.458 - 0.888i)T \)
71 \( 1 + (-0.0285 - 0.999i)T \)
73 \( 1 + (-0.956 - 0.290i)T \)
79 \( 1 + (0.345 + 0.938i)T \)
83 \( 1 + (0.856 + 0.516i)T \)
89 \( 1 + (-0.723 - 0.690i)T \)
97 \( 1 + (-0.441 + 0.897i)T \)
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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

−17.99477779560066665874198344017, −17.38752901941944033493215229492, −17.25000113228663835020983167334, −16.23827868629014685868963969570, −15.576358750145916981175535331893, −14.74111234185601611774994752347, −13.701292873343335564214295996995, −13.06436584471682067764037792462, −12.5150835951342061243908758024, −11.96178949783743898810371092272, −11.18833379791451165248262682567, −10.57492735214855043408730544319, −10.1439613094436091480161853592, −9.40666138551258958385918504117, −8.29803849786006701750007018981, −7.94726019454335533507198670639, −6.95379551958319438776771842436, −6.04509014101848549497657709768, −5.41671690401549878809174657698, −4.528657454918080554533266301470, −3.87536778857186296146182716378, −3.038668730041125605428655011649, −1.95429417574152338700352170027, −1.30976501218056281735201346937, −0.301965320533994019174755432973, 0.77735532235088482992097812786, 1.63851895159757777406392001012, 2.87129074657865533389746547867, 4.301923428905894975540743818073, 4.584737297501508483730279307499, 5.28938688902412033267120551419, 6.34945982686580878846041414178, 6.57909792616480874797074012189, 7.27464704125793757708107188658, 8.18853224011142935127972274254, 9.04610361431217572377277265815, 9.52479660434192623111377241901, 10.4541371593274388982001482183, 10.90941017158206039860921429513, 11.79884542322790860435143528676, 12.45960686622711263695632348488, 13.37690610006087105366697585703, 13.90898691309308208946411639378, 14.78544330944677180078781400302, 15.42780471268263410906494538935, 16.17469365259421890264352917548, 16.51702132923739261657802538426, 17.23149194500471792976760823327, 17.79894878875947728297929254003, 18.47040944499904641781080841635

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