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Properties

Label 1-4235-4235.1104-r1-0-0
Degree $1$
Conductor $4235$
Sign $-0.969 + 0.245i$
Analytic cond. $455.113$
Root an. cond. $455.113$
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  + (−0.640 − 0.768i)2-s + (0.913 − 0.406i)3-s + (−0.179 + 0.983i)4-s + (−0.897 − 0.441i)6-s + (0.870 − 0.491i)8-s + (0.669 − 0.743i)9-s + (0.235 + 0.971i)12-s + (−0.0285 − 0.999i)13-s + (−0.935 − 0.353i)16-s + (−0.999 + 0.0190i)17-s + (−0.999 − 0.0380i)18-s + (−0.820 + 0.572i)19-s + (−0.0475 + 0.998i)23-s + (0.595 − 0.803i)24-s + (−0.749 + 0.662i)26-s + (0.309 − 0.951i)27-s + ⋯
L(s)  = 1  + (−0.640 − 0.768i)2-s + (0.913 − 0.406i)3-s + (−0.179 + 0.983i)4-s + (−0.897 − 0.441i)6-s + (0.870 − 0.491i)8-s + (0.669 − 0.743i)9-s + (0.235 + 0.971i)12-s + (−0.0285 − 0.999i)13-s + (−0.935 − 0.353i)16-s + (−0.999 + 0.0190i)17-s + (−0.999 − 0.0380i)18-s + (−0.820 + 0.572i)19-s + (−0.0475 + 0.998i)23-s + (0.595 − 0.803i)24-s + (−0.749 + 0.662i)26-s + (0.309 − 0.951i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.969 + 0.245i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.969 + 0.245i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.969 + 0.245i$
Analytic conductor: \(455.113\)
Root analytic conductor: \(455.113\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (1:\ ),\ -0.969 + 0.245i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1317160249 - 1.057147585i\)
\(L(\frac12)\) \(\approx\) \(-0.1317160249 - 1.057147585i\)
\(L(1)\) \(\approx\) \(0.8254192913 - 0.4641745546i\)
\(L(1)\) \(\approx\) \(0.8254192913 - 0.4641745546i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.640 - 0.768i)T \)
3 \( 1 + (0.913 - 0.406i)T \)
13 \( 1 + (-0.0285 - 0.999i)T \)
17 \( 1 + (-0.999 + 0.0190i)T \)
19 \( 1 + (-0.820 + 0.572i)T \)
23 \( 1 + (-0.0475 + 0.998i)T \)
29 \( 1 + (0.974 + 0.226i)T \)
31 \( 1 + (-0.997 + 0.0760i)T \)
37 \( 1 + (0.991 + 0.132i)T \)
41 \( 1 + (-0.696 + 0.717i)T \)
43 \( 1 + (-0.415 + 0.909i)T \)
47 \( 1 + (0.999 - 0.0380i)T \)
53 \( 1 + (0.935 - 0.353i)T \)
59 \( 1 + (0.969 - 0.244i)T \)
61 \( 1 + (-0.345 - 0.938i)T \)
67 \( 1 + (0.786 + 0.618i)T \)
71 \( 1 + (-0.921 + 0.389i)T \)
73 \( 1 + (0.449 - 0.893i)T \)
79 \( 1 + (0.953 + 0.299i)T \)
83 \( 1 + (-0.254 - 0.967i)T \)
89 \( 1 + (-0.981 - 0.189i)T \)
97 \( 1 + (0.993 + 0.113i)T \)
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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

−18.53382062726338890213575289811, −17.997527307641433515832986618100, −16.95908638153867717941955394737, −16.59815867233182313291461259722, −15.778459454490268436907473768888, −15.25225464145852089101782965120, −14.68453192274622104126029634958, −13.95669286896483116098134128647, −13.49975282931452155748852036738, −12.64360998392506729069308651423, −11.50159130151167588331468000962, −10.71158526166642380494459251248, −10.21687387035458479764560956535, −9.339266244190200212045275742761, −8.73155307618989127668835327451, −8.510270593316615092563015191115, −7.40560383052717994078760513428, −6.89269579338410482324499262416, −6.20658041832690245988532526898, −5.10337859418260275338756390264, −4.415073805786969874893588182199, −3.888607147097652057278277656682, −2.3590380258765331481611381393, −2.17882329768215500451688446798, −0.910869526275310672943887262503, 0.18653887699691109462334174134, 1.12474979278327596171960683794, 1.90739175296802388674489201864, 2.60337939838285651996561052021, 3.33713517968218932854132188675, 3.993349080838870237610083894314, 4.85221677006701812313210397041, 6.09075885419817860800250692246, 6.941839901246523868387947748891, 7.63461232628628084077870793039, 8.31746216018076934023517973392, 8.73679751734439660231722659650, 9.60857295754890046605524807581, 10.13280107413317493295267211146, 10.91540438457712034451776335212, 11.662650637444555839426492126681, 12.497479378892389147029909862481, 13.038575373517068574969674710567, 13.46575908164214648959838546061, 14.385490758336914633896888330387, 15.14164074174681465519417707115, 15.77275433693769075935224528268, 16.63024661103139575212807894669, 17.483514401570863188025674806776, 18.02677264917703902943800988501

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