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Properties

Label 1-4235-4235.2-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.769 - 0.638i$
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  + (0.893 − 0.449i)2-s + (0.743 − 0.669i)3-s + (0.595 − 0.803i)4-s + (0.362 − 0.931i)6-s + (0.170 − 0.985i)8-s + (0.104 − 0.994i)9-s + (−0.0950 − 0.995i)12-s + (−0.491 − 0.870i)13-s + (−0.290 − 0.956i)16-s + (0.983 + 0.179i)17-s + (−0.353 − 0.935i)18-s + (0.879 − 0.475i)19-s + (0.945 + 0.327i)23-s + (−0.532 − 0.846i)24-s + (−0.830 − 0.556i)26-s + (−0.587 − 0.809i)27-s + ⋯
L(s)  = 1  + (0.893 − 0.449i)2-s + (0.743 − 0.669i)3-s + (0.595 − 0.803i)4-s + (0.362 − 0.931i)6-s + (0.170 − 0.985i)8-s + (0.104 − 0.994i)9-s + (−0.0950 − 0.995i)12-s + (−0.491 − 0.870i)13-s + (−0.290 − 0.956i)16-s + (0.983 + 0.179i)17-s + (−0.353 − 0.935i)18-s + (0.879 − 0.475i)19-s + (0.945 + 0.327i)23-s + (−0.532 − 0.846i)24-s + (−0.830 − 0.556i)26-s + (−0.587 − 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.769 - 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.769 - 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.769 - 0.638i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.769 - 0.638i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.534976839 - 4.256258839i\)
\(L(\frac12)\) \(\approx\) \(1.534976839 - 4.256258839i\)
\(L(1)\) \(\approx\) \(1.822310169 - 1.545215159i\)
\(L(1)\) \(\approx\) \(1.822310169 - 1.545215159i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.893 - 0.449i)T \)
3 \( 1 + (0.743 - 0.669i)T \)
13 \( 1 + (-0.491 - 0.870i)T \)
17 \( 1 + (0.983 + 0.179i)T \)
19 \( 1 + (0.879 - 0.475i)T \)
23 \( 1 + (0.945 + 0.327i)T \)
29 \( 1 + (-0.564 - 0.825i)T \)
31 \( 1 + (0.749 + 0.662i)T \)
37 \( 1 + (-0.299 + 0.953i)T \)
41 \( 1 + (0.254 + 0.967i)T \)
43 \( 1 + (0.989 - 0.142i)T \)
47 \( 1 + (0.353 - 0.935i)T \)
53 \( 1 + (0.956 + 0.290i)T \)
59 \( 1 + (0.710 + 0.703i)T \)
61 \( 1 + (-0.548 - 0.836i)T \)
67 \( 1 + (-0.998 + 0.0475i)T \)
71 \( 1 + (0.610 - 0.791i)T \)
73 \( 1 + (-0.875 + 0.483i)T \)
79 \( 1 + (-0.969 - 0.244i)T \)
83 \( 1 + (0.996 + 0.0855i)T \)
89 \( 1 + (-0.235 - 0.971i)T \)
97 \( 1 + (-0.884 - 0.466i)T \)
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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

−18.94394384329989559681021959219, −17.76444740091700291132806154640, −16.94178844256222466722798120648, −16.317610271623935608183312419750, −16.001226746009021889138240121427, −15.014157632681419708760152779533, −14.59646440003470087448570784278, −14.037316669510837222379386916100, −13.48697113789442708081372425961, −12.567469404459887368599122748811, −12.00853053242462114864744284925, −11.145957332891130545973399164618, −10.48016011321712409518430989198, −9.51146267700099099762627238358, −9.009095261064438106911989696660, −8.096462847327700803191189280100, −7.41374874293096047069916443101, −6.94143049356626929627771911637, −5.680623009354107348815920473967, −5.29053264963284439547621815850, −4.35152169131008455518525931189, −3.86470692018634304796068742799, −2.99531536671034606903551136447, −2.44189965717755214687638358531, −1.41118538865417382816973125976, 0.823388942658332244209980285821, 1.396111963109411770334653636165, 2.51254194663472330021246419377, 3.00042029317810041594826221132, 3.60764136810496629797922318675, 4.60886159709847051997705221967, 5.42578634736743306813037353623, 6.048792346556628854204250194040, 7.040921274388483654581479936840, 7.47346280800846118824205057299, 8.28530984650653295346389798490, 9.28612899145993931428275216520, 9.89527279459297257235201398962, 10.57701251822663412727445640251, 11.62678101692120096169113194329, 12.05530986738318435584787345680, 12.76798751263391752364381879516, 13.40095827471120002082715317151, 13.8431565257126748543111254645, 14.689293759262406252342194684835, 15.13542378441779674981500603222, 15.71028573815752105871185458894, 16.73814920046938482490369442303, 17.56079136311942305961832941037, 18.33320713278335204052827928803

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