Properties

Label 1-4235-4235.1263-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.789 + 0.614i$
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  + (0.956 + 0.290i)2-s + (0.207 + 0.978i)3-s + (0.830 + 0.556i)4-s + (−0.0855 + 0.996i)6-s + (0.633 + 0.774i)8-s + (−0.913 + 0.406i)9-s + (−0.371 + 0.928i)12-s + (0.884 + 0.466i)13-s + (0.380 + 0.924i)16-s + (0.662 + 0.749i)17-s + (−0.992 + 0.123i)18-s + (−0.398 + 0.917i)19-s + (0.971 + 0.235i)23-s + (−0.625 + 0.780i)24-s + (0.710 + 0.703i)26-s + (−0.587 − 0.809i)27-s + ⋯
L(s)  = 1  + (0.956 + 0.290i)2-s + (0.207 + 0.978i)3-s + (0.830 + 0.556i)4-s + (−0.0855 + 0.996i)6-s + (0.633 + 0.774i)8-s + (−0.913 + 0.406i)9-s + (−0.371 + 0.928i)12-s + (0.884 + 0.466i)13-s + (0.380 + 0.924i)16-s + (0.662 + 0.749i)17-s + (−0.992 + 0.123i)18-s + (−0.398 + 0.917i)19-s + (0.971 + 0.235i)23-s + (−0.625 + 0.780i)24-s + (0.710 + 0.703i)26-s + (−0.587 − 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.789 + 0.614i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.789 + 0.614i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.789 + 0.614i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.789 + 0.614i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.301411116 + 3.791043631i\)
\(L(\frac12)\) \(\approx\) \(1.301411116 + 3.791043631i\)
\(L(1)\) \(\approx\) \(1.669851283 + 1.381870254i\)
\(L(1)\) \(\approx\) \(1.669851283 + 1.381870254i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.956 + 0.290i)T \)
3 \( 1 + (0.207 + 0.978i)T \)
13 \( 1 + (0.884 + 0.466i)T \)
17 \( 1 + (0.662 + 0.749i)T \)
19 \( 1 + (-0.398 + 0.917i)T \)
23 \( 1 + (0.971 + 0.235i)T \)
29 \( 1 + (0.736 - 0.676i)T \)
31 \( 1 + (0.969 + 0.244i)T \)
37 \( 1 + (-0.938 - 0.345i)T \)
41 \( 1 + (-0.516 - 0.856i)T \)
43 \( 1 + (-0.540 + 0.841i)T \)
47 \( 1 + (0.992 + 0.123i)T \)
53 \( 1 + (0.924 + 0.380i)T \)
59 \( 1 + (-0.999 - 0.0190i)T \)
61 \( 1 + (0.683 - 0.730i)T \)
67 \( 1 + (0.189 - 0.981i)T \)
71 \( 1 + (-0.870 - 0.491i)T \)
73 \( 1 + (0.901 + 0.432i)T \)
79 \( 1 + (-0.548 + 0.836i)T \)
83 \( 1 + (-0.336 - 0.941i)T \)
89 \( 1 + (0.580 + 0.814i)T \)
97 \( 1 + (0.931 - 0.362i)T \)
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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

−18.352338731094389211451762503586, −17.45917904924954576886793635745, −16.75543822143001068534104183548, −15.85660652534463637386057924174, −15.23427351795088244813492427868, −14.56640388316859757271584123922, −13.77077203352738425107576021417, −13.40120173465154834847060696333, −12.80048292801731868119859143806, −12.00035173036767798877856636729, −11.57987466289184277044043878791, −10.74897468808682838136043748705, −10.09711721566629725928150253771, −8.94851933035179894055077217925, −8.3850893704337069279964187204, −7.36449723644211710620182101224, −6.85446354496541364416595455436, −6.21569880182487584074308810544, −5.38480538871088412526173053291, −4.783821470093207125256287290592, −3.65867914143880623644740052359, −2.96835435809440585276211435599, −2.4523015980310464358545236149, −1.33472412952361701280478127593, −0.78463306965475835823284620438, 1.36433334535245249602901063861, 2.3178091893697934133930451605, 3.26793712496388731039582143788, 3.74270907769068204396607345026, 4.43527853487697256150659513802, 5.17442052574802154620449128862, 5.93415579844446773289096587546, 6.4579675837103781443820776521, 7.50046862692370400958790470825, 8.35134207335838530951273569402, 8.737067380400892083088393010659, 9.87177043377897444458786361742, 10.54160920309520786272338229134, 11.08202142087920211856001949407, 11.92425003286476305582898742578, 12.47941009811856270702837760544, 13.48286846774832137185220939969, 13.98101205692012523645532079069, 14.5532229569147699627966430855, 15.4222711271087911353401992193, 15.59962724353859025243663990322, 16.58445086270818375548864055767, 16.90711913815539510253843121840, 17.634380109619926029918504317869, 18.91140559928668385354895103928

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