Properties

Label 1-4235-4235.157-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.977 - 0.210i$
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  + (0.299 + 0.953i)2-s + (−0.406 + 0.913i)3-s + (−0.820 + 0.572i)4-s + (−0.993 − 0.113i)6-s + (−0.791 − 0.610i)8-s + (−0.669 − 0.743i)9-s + (−0.189 − 0.981i)12-s + (−0.389 − 0.921i)13-s + (0.345 − 0.938i)16-s + (−0.263 − 0.964i)17-s + (0.508 − 0.861i)18-s + (−0.625 − 0.780i)19-s + (−0.618 + 0.786i)23-s + (0.879 − 0.475i)24-s + (0.761 − 0.647i)26-s + (0.951 − 0.309i)27-s + ⋯
L(s)  = 1  + (0.299 + 0.953i)2-s + (−0.406 + 0.913i)3-s + (−0.820 + 0.572i)4-s + (−0.993 − 0.113i)6-s + (−0.791 − 0.610i)8-s + (−0.669 − 0.743i)9-s + (−0.189 − 0.981i)12-s + (−0.389 − 0.921i)13-s + (0.345 − 0.938i)16-s + (−0.263 − 0.964i)17-s + (0.508 − 0.861i)18-s + (−0.625 − 0.780i)19-s + (−0.618 + 0.786i)23-s + (0.879 − 0.475i)24-s + (0.761 − 0.647i)26-s + (0.951 − 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.210i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.210i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.977 - 0.210i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.977 - 0.210i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.07137120767 + 0.6704233615i\)
\(L(\frac12)\) \(\approx\) \(-0.07137120767 + 0.6704233615i\)
\(L(1)\) \(\approx\) \(0.5924003122 + 0.5225626779i\)
\(L(1)\) \(\approx\) \(0.5924003122 + 0.5225626779i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.299 - 0.953i)T \)
3 \( 1 + (0.406 - 0.913i)T \)
13 \( 1 + (0.389 + 0.921i)T \)
17 \( 1 + (0.263 + 0.964i)T \)
19 \( 1 + (0.625 + 0.780i)T \)
23 \( 1 + (0.618 - 0.786i)T \)
29 \( 1 + (-0.998 - 0.0570i)T \)
31 \( 1 + (0.483 - 0.875i)T \)
37 \( 1 + (0.956 + 0.290i)T \)
41 \( 1 + (0.198 + 0.980i)T \)
43 \( 1 + (-0.281 - 0.959i)T \)
47 \( 1 + (0.508 + 0.861i)T \)
53 \( 1 + (-0.938 + 0.345i)T \)
59 \( 1 + (0.948 - 0.318i)T \)
61 \( 1 + (-0.217 - 0.976i)T \)
67 \( 1 + (0.0950 + 0.995i)T \)
71 \( 1 + (-0.774 - 0.633i)T \)
73 \( 1 + (-0.244 + 0.969i)T \)
79 \( 1 + (-0.432 - 0.901i)T \)
83 \( 1 + (-0.441 - 0.897i)T \)
89 \( 1 + (0.888 - 0.458i)T \)
97 \( 1 + (-0.999 - 0.0285i)T \)
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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

−18.38497172002120663041277842624, −17.28416233740188811411248460301, −17.077830006607752627886043618941, −16.08995780317161713635857123927, −15.010978214351722308318445945896, −14.31288386383608170325256581467, −13.8784202816108584506215884198, −13.00161559589659019520018283485, −12.520896212397877705854581838517, −11.94927074109858948987378427039, −11.34782077007642334651913142413, −10.54180489146094364176805909545, −10.06866832819940743540208145732, −8.98871665680585093893498385686, −8.39340428402432042425091066285, −7.66707123302254328899289417059, −6.44529323399747002001172894978, −6.24664690164560961455368855971, −5.23503016406762647310458627918, −4.4570362783221997757880179144, −3.77905339020792219115117384950, −2.672775237632797980989391339197, −1.97581166769379516611144858628, −1.46476908221919174643820336079, −0.24482033706623887938421656062, 0.74281431327563441765405285416, 2.48201456625235654128930211690, 3.308382038258130891345182779808, 3.967489575259809224742715238093, 5.01737382543093897933255037215, 5.09893711020495045129971661887, 6.0478905651766534769857786030, 6.79664762157162789717540946894, 7.473099504802431833205400301335, 8.44234555196322847733508592520, 8.9689698622800623612592951305, 9.73243938659307765610391111188, 10.371618519787690576462212622209, 11.18434833654929675726146412614, 12.079873942154655172270424015761, 12.54607726542418490468202259511, 13.59941053525153462230557681774, 14.04299115032915447810678068036, 14.98166858011921515801660636560, 15.36950031080301357925955817634, 15.98981997983691254889277035405, 16.512789668540174703704079433500, 17.37913759738551193288900681357, 17.76522424944646229482073966412, 18.284855207915664719673662620372

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