↑

Properties

Label 1-4235-4235.2909-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.943 - 0.332i$
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

Related objects

Downloads

Learn more

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

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

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.943 - 0.332i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (2909, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.943 - 0.332i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04356707821 + 0.2545205501i\)
\(L(\frac12)\) \(\approx\) \(-0.04356707821 + 0.2545205501i\)
\(L(1)\) \(\approx\) \(0.4670406096 + 0.1629335689i\)
\(L(1)\) \(\approx\) \(0.4670406096 + 0.1629335689i\)

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 \( 1 + (-0.198 - 0.980i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.92535943882164890962789220518, −17.457272051938418231258969986092, −16.74645294918741404516914172149, −16.35537222997367806608669740974, −15.41144888016310681947076719, −14.81047369295335031973543547123, −13.868953641102410320704359589764, −13.06053500411774957004527311091, −12.33086701917483121322309462927, −11.77078767817384742055909113644, −11.09106593996378715368092056004, −10.57560354079172842121839142282, −9.53156112738712907305169056407, −9.23503883669884507708301142841, −8.041821851704465437960005200654, −7.4896368853165668913107662453, −7.0923521986523469452353583003, −6.27450326692901924288930435274, −5.35818815439606643216903273417, −4.989256019651650346752148746686, −3.417173843945940526625086482456, −2.62315787701454911589488803403, −1.866743647973515968643590504040, −0.95200799902512157776331753927, −0.13877017138754056516519585677, 1.09816443596433065567468369121, 1.91337889131549420047990278943, 3.08159948597141892979659710440, 3.65496542509481086935313143815, 4.73653625305420289988876844735, 5.50477473060168015391242788320, 6.17563281845975701543963053315, 6.97474639136863301233310186560, 7.664847911630616419783721453450, 8.50232442043079017436025403697, 9.31364222411621615450649471090, 9.82020495104913682795891789082, 10.39184380350560430392411689643, 11.093859079807010878169264752457, 11.73102593734534272960676822185, 12.34298669776885147626211450470, 13.00941921131343921963360416316, 14.548575571359584695718589157760, 14.74876299215641038851860507158, 15.49732290065575185232655796952, 16.41452387258966661878461336949, 16.86357808516095626399048162813, 17.04217654864318503254693088068, 18.21408284065444933894611077296, 18.473131530143300641691022634822

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