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Properties

Label 1-4235-4235.1073-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.264 + 0.964i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.264 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.264 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.264 + 0.964i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1073, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.264 + 0.964i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3062616073 - 0.4016743016i\)
\(L(\frac12)\) \(\approx\) \(-0.3062616073 - 0.4016743016i\)
\(L(1)\) \(\approx\) \(1.200646053 - 0.7944398443i\)
\(L(1)\) \(\approx\) \(1.200646053 - 0.7944398443i\)

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

−19.01519923729185199128635844998, −17.9554623484092448345056619640, −17.24199990922760995014337464314, −16.478027232941050621286910623550, −16.14980073730982865571306706925, −15.35410774000889453233945590322, −14.65772646330890774551502028196, −14.42739193591934189371469873260, −13.51749945188405579110811761694, −12.83405141173477860318699815518, −12.0557738741940359281049633963, −11.40212586254769827574455843755, −10.68684593026632439190556514545, −10.025861214684287560199276882042, −9.25308058648411272041505315154, −8.351531893757905215686168918265, −7.72044370097866105287041965012, −6.940999782908613835766595792390, −5.959408143202725287992477082712, −5.408564686789009139520473476577, −4.6793796175104656834730472305, −4.0554555212342183349502204162, −3.34058592096400593063231189137, −2.55281354454224524916779608514, −1.863528729422272557202512973952, 0.07886252125364524737008949926, 1.37221507957790357465290386980, 2.18405000825938525120994027665, 2.54241938715241339775234002838, 3.666373910494722872575750277730, 4.296184957440981576285941094601, 5.24367230502858434560877281901, 5.97700365240067874652811389750, 6.64213239929432366713439692879, 7.249655875472573905939873779711, 7.928346164544288836367456336603, 8.90514419502130835022312206615, 9.646742939604087972399206712463, 10.58669142060437705247971150754, 11.355898957525341610527945602954, 11.852480468358612744341606934304, 12.570215822153419930015758383, 13.18277572562703726717875618980, 13.60499965906299273301725016408, 14.5103344850931489926093500703, 14.867538835432056084057504397084, 15.621180348324967272621765438716, 16.57669014755739838417843278437, 17.22511992222818798793961201114, 17.94879524763558110265321495298

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