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Properties

Label 1-4029-4029.23-r0-0-0
Degree $1$
Conductor $4029$
Sign $0.173 - 0.984i$
Analytic cond. $18.7105$
Root an. cond. $18.7105$
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  + (0.965 − 0.258i)2-s + (0.866 − 0.5i)4-s + (0.608 − 0.793i)5-s + (0.991 − 0.130i)7-s + (0.707 − 0.707i)8-s + (0.382 − 0.923i)10-s + (−0.130 − 0.991i)11-s + (0.866 + 0.5i)13-s + (0.923 − 0.382i)14-s + (0.5 − 0.866i)16-s + (0.258 − 0.965i)19-s + (0.130 − 0.991i)20-s + (−0.382 − 0.923i)22-s + (0.130 + 0.991i)23-s + (−0.258 − 0.965i)25-s + (0.965 + 0.258i)26-s + ⋯
L(s)  = 1  + (0.965 − 0.258i)2-s + (0.866 − 0.5i)4-s + (0.608 − 0.793i)5-s + (0.991 − 0.130i)7-s + (0.707 − 0.707i)8-s + (0.382 − 0.923i)10-s + (−0.130 − 0.991i)11-s + (0.866 + 0.5i)13-s + (0.923 − 0.382i)14-s + (0.5 − 0.866i)16-s + (0.258 − 0.965i)19-s + (0.130 − 0.991i)20-s + (−0.382 − 0.923i)22-s + (0.130 + 0.991i)23-s + (−0.258 − 0.965i)25-s + (0.965 + 0.258i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4029\)    =    \(3 \cdot 17 \cdot 79\)
Sign: $0.173 - 0.984i$
Analytic conductor: \(18.7105\)
Root analytic conductor: \(18.7105\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4029} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4029,\ (0:\ ),\ 0.173 - 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.860329075 - 3.239501925i\)
\(L(\frac12)\) \(\approx\) \(3.860329075 - 3.239501925i\)
\(L(1)\) \(\approx\) \(2.341160552 - 1.023437291i\)
\(L(1)\) \(\approx\) \(2.341160552 - 1.023437291i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
79 \( 1 \)
good2 \( 1 + (0.965 - 0.258i)T \)
5 \( 1 + (0.608 - 0.793i)T \)
7 \( 1 + (0.991 - 0.130i)T \)
11 \( 1 + (-0.130 - 0.991i)T \)
13 \( 1 + (0.866 + 0.5i)T \)
19 \( 1 + (0.258 - 0.965i)T \)
23 \( 1 + (0.130 + 0.991i)T \)
29 \( 1 + (-0.608 + 0.793i)T \)
31 \( 1 + (0.793 + 0.608i)T \)
37 \( 1 + (-0.130 + 0.991i)T \)
41 \( 1 + (0.382 - 0.923i)T \)
43 \( 1 + (0.258 + 0.965i)T \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 + (-0.258 + 0.965i)T \)
59 \( 1 + (0.258 + 0.965i)T \)
61 \( 1 + (0.382 - 0.923i)T \)
67 \( 1 - T \)
71 \( 1 + (-0.923 + 0.382i)T \)
73 \( 1 + (-0.608 + 0.793i)T \)
83 \( 1 + (0.258 - 0.965i)T \)
89 \( 1 - iT \)
97 \( 1 + (0.382 + 0.923i)T \)
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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

−18.477062229154763022643043310333, −17.79451346791892630005910170535, −17.370979934781556499281719701235, −16.49354707269321630791578793380, −15.6640761647492481523329794699, −14.91911764245593747258304175946, −14.67646859303599040282808028398, −13.90366717427151092422344612263, −13.31847775039232033636342707326, −12.54084670399433879425375744846, −11.84709500632295657776349196241, −11.091202148873535427362800870112, −10.54987787114958380798218628083, −9.86211888525981492227443116708, −8.72977535409559332801817015769, −7.831292659491887980092415482596, −7.44531596349119440357898749324, −6.47042676572044147930372771624, −5.88050121763260772247788855998, −5.27111381282252239130623766801, −4.3615295757390709401452472281, −3.75439639288321624850750092817, −2.68942166155389904387450184660, −2.14281693719562214488600586982, −1.36358662811162810177115062890, 1.10459284808616441683552832618, 1.38025018451381847805286092173, 2.42848886244926174266425081124, 3.28117650464811744998889918978, 4.180889394804623793167556734739, 4.8325730252776912858058682919, 5.49971489659421505451245293449, 6.02069998891044532881285549989, 6.94174922027402557663963001748, 7.78494009882691020911050818156, 8.71655363827553705607496052738, 9.17533360681785121455276127270, 10.308381088413341657742029994259, 10.91921937822062674311901444543, 11.56055926923994930072933049216, 12.0705481613512514440379437757, 13.16419754889753375033431527717, 13.50067243312523752608514433972, 14.01302405405026433404161068677, 14.69692260500268171511005899347, 15.73121915885210531175310785302, 16.021724321539601034633350565458, 16.94269164178202937500715701272, 17.49496992691470814998476328432, 18.38245201859816719971533170555

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