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Properties

Label 1-4235-4235.754-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.565 - 0.824i$
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  + (0.00951 + 0.999i)2-s + (−0.978 − 0.207i)3-s + (−0.999 + 0.0190i)4-s + (0.198 − 0.980i)6-s + (−0.0285 − 0.999i)8-s + (0.913 + 0.406i)9-s + (0.981 + 0.189i)12-s + (−0.0855 + 0.996i)13-s + (0.999 − 0.0380i)16-s + (−0.548 + 0.836i)17-s + (−0.398 + 0.917i)18-s + (0.964 − 0.263i)19-s + (0.786 − 0.618i)23-s + (−0.179 + 0.983i)24-s + (−0.997 − 0.0760i)26-s + (−0.809 − 0.587i)27-s + ⋯
L(s)  = 1  + (0.00951 + 0.999i)2-s + (−0.978 − 0.207i)3-s + (−0.999 + 0.0190i)4-s + (0.198 − 0.980i)6-s + (−0.0285 − 0.999i)8-s + (0.913 + 0.406i)9-s + (0.981 + 0.189i)12-s + (−0.0855 + 0.996i)13-s + (0.999 − 0.0380i)16-s + (−0.548 + 0.836i)17-s + (−0.398 + 0.917i)18-s + (0.964 − 0.263i)19-s + (0.786 − 0.618i)23-s + (−0.179 + 0.983i)24-s + (−0.997 − 0.0760i)26-s + (−0.809 − 0.587i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.565 - 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.565 - 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.565 - 0.824i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (754, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.565 - 0.824i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.09405394794 + 0.1785244168i\)
\(L(\frac12)\) \(\approx\) \(-0.09405394794 + 0.1785244168i\)
\(L(1)\) \(\approx\) \(0.5453553236 + 0.3077279952i\)
\(L(1)\) \(\approx\) \(0.5453553236 + 0.3077279952i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.00951 + 0.999i)T \)
3 \( 1 + (-0.978 - 0.207i)T \)
13 \( 1 + (-0.0855 + 0.996i)T \)
17 \( 1 + (-0.548 + 0.836i)T \)
19 \( 1 + (0.964 - 0.263i)T \)
23 \( 1 + (0.786 - 0.618i)T \)
29 \( 1 + (-0.774 + 0.633i)T \)
31 \( 1 + (0.683 - 0.730i)T \)
37 \( 1 + (-0.797 - 0.603i)T \)
41 \( 1 + (-0.736 + 0.676i)T \)
43 \( 1 + (-0.959 - 0.281i)T \)
47 \( 1 + (-0.398 - 0.917i)T \)
53 \( 1 + (-0.999 - 0.0380i)T \)
59 \( 1 + (-0.953 + 0.299i)T \)
61 \( 1 + (0.861 + 0.508i)T \)
67 \( 1 + (0.995 + 0.0950i)T \)
71 \( 1 + (-0.362 - 0.931i)T \)
73 \( 1 + (-0.640 + 0.768i)T \)
79 \( 1 + (0.991 + 0.132i)T \)
83 \( 1 + (-0.696 + 0.717i)T \)
89 \( 1 + (0.888 + 0.458i)T \)
97 \( 1 + (0.941 - 0.336i)T \)
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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

−17.74300972985332975574895313098, −17.587772589532922294135630845033, −16.820285274763454555242260799578, −15.83893144732073431400926365656, −15.38980557508289428398731853136, −14.43275683514550694587682887708, −13.522000422193677485480840339761, −13.092074206600389765801856091828, −12.24560968766055897156273652402, −11.738242282473382213013898364637, −11.128260785028867038705103332745, −10.51384728252310553062631771055, −9.76765107454929734871310582314, −9.35971731526697790154265852789, −8.3185895920765406553761393409, −7.51358233808297771949171904834, −6.66142439974331077803903464551, −5.65166050914113173153660129215, −5.09718642198733623364795655238, −4.592839878383033922745273071979, −3.45729116909879184984372270001, −3.044196854556953250400775229598, −1.79952820245446321312268813965, −1.032932997530320585423122592075, −0.08162158706242329253405130049, 1.10174515983261345724807870336, 1.97981983818597304711238287854, 3.41090305808390981740589988596, 4.24025160000278892075671767102, 4.93501102580843588154240917242, 5.47601085059476714245755625600, 6.44188803383458702206047574990, 6.75969252715231210934567845908, 7.465364048467861446975612416712, 8.32300122270250692713426772614, 9.091044798039897577588438116489, 9.79474709514735417198152070367, 10.56288427774693625652929594646, 11.369493435298981734252233752, 12.03454772902191833889525932795, 12.860658029693656177243145687091, 13.36441871354141982847414353217, 14.08785301166201821535540768521, 14.95546212664158023298436303944, 15.499835124765968623175863909402, 16.31445383205268126124694244984, 16.75229018171248915555768864060, 17.306576656945015129348868806, 17.97920795378909565119241386359, 18.65744745021653749441510595290

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