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Properties

Label 1-4029-4029.140-r0-0-0
Degree $1$
Conductor $4029$
Sign $0.853 - 0.521i$
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  + (−0.354 + 0.935i)2-s + (−0.748 − 0.663i)4-s + (0.992 + 0.120i)5-s + (0.464 − 0.885i)7-s + (0.885 − 0.464i)8-s + (−0.464 + 0.885i)10-s + (0.992 − 0.120i)11-s + (−0.748 − 0.663i)13-s + (0.663 + 0.748i)14-s + (0.120 + 0.992i)16-s + (0.970 − 0.239i)19-s + (−0.663 − 0.748i)20-s + (−0.239 + 0.970i)22-s − i·23-s + (0.970 + 0.239i)25-s + (0.885 − 0.464i)26-s + ⋯
L(s)  = 1  + (−0.354 + 0.935i)2-s + (−0.748 − 0.663i)4-s + (0.992 + 0.120i)5-s + (0.464 − 0.885i)7-s + (0.885 − 0.464i)8-s + (−0.464 + 0.885i)10-s + (0.992 − 0.120i)11-s + (−0.748 − 0.663i)13-s + (0.663 + 0.748i)14-s + (0.120 + 0.992i)16-s + (0.970 − 0.239i)19-s + (−0.663 − 0.748i)20-s + (−0.239 + 0.970i)22-s − i·23-s + (0.970 + 0.239i)25-s + (0.885 − 0.464i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.853 - 0.521i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.853 - 0.521i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4029\)    =    \(3 \cdot 17 \cdot 79\)
Sign: $0.853 - 0.521i$
Analytic conductor: \(18.7105\)
Root analytic conductor: \(18.7105\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4029} (140, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4029,\ (0:\ ),\ 0.853 - 0.521i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.746259851 - 0.4912163194i\)
\(L(\frac12)\) \(\approx\) \(1.746259851 - 0.4912163194i\)
\(L(1)\) \(\approx\) \(1.114345359 + 0.1338267838i\)
\(L(1)\) \(\approx\) \(1.114345359 + 0.1338267838i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
79 \( 1 \)
good2 \( 1 + (-0.354 + 0.935i)T \)
5 \( 1 + (0.992 + 0.120i)T \)
7 \( 1 + (0.464 - 0.885i)T \)
11 \( 1 + (0.992 - 0.120i)T \)
13 \( 1 + (-0.748 - 0.663i)T \)
19 \( 1 + (0.970 - 0.239i)T \)
23 \( 1 - iT \)
29 \( 1 + (-0.822 - 0.568i)T \)
31 \( 1 + (0.935 + 0.354i)T \)
37 \( 1 + (-0.239 - 0.970i)T \)
41 \( 1 + (0.992 + 0.120i)T \)
43 \( 1 + (0.120 - 0.992i)T \)
47 \( 1 + (-0.970 - 0.239i)T \)
53 \( 1 + (-0.885 + 0.464i)T \)
59 \( 1 + (0.748 - 0.663i)T \)
61 \( 1 + (0.239 + 0.970i)T \)
67 \( 1 + (-0.354 - 0.935i)T \)
71 \( 1 + (-0.464 - 0.885i)T \)
73 \( 1 + (0.663 + 0.748i)T \)
83 \( 1 + (-0.748 - 0.663i)T \)
89 \( 1 + (-0.885 - 0.464i)T \)
97 \( 1 + (0.239 + 0.970i)T \)
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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

−18.535135093965694275662695341448, −17.83874524466338902285873152967, −17.421386523420830324349136557275, −16.77545439110971080291604606156, −16.02415554164943090946677326986, −14.83777993134752345706949843651, −14.31224753088117144288129018934, −13.74138516916114428814323448940, −12.88887982465755731429528174111, −12.26300446250210308631065095961, −11.53643363385241747410836580626, −11.233880503518203553568336039904, −9.909228183184217443978432452179, −9.63503523028420820159955195360, −9.10604209851218916073079177707, −8.34291884993476514768449902315, −7.48633910589763002542261280649, −6.57897523964698389413463127674, −5.63300365807347946068796052663, −4.993958433751644332339294877, −4.26150319476555029535858335886, −3.19341253433439576827618451881, −2.50825778332897589305430808612, −1.62097820389784255625963032677, −1.29653489191358303140286277133, 0.60839476735504394982942206366, 1.3394529756689631520908877508, 2.29943632878787655717619179287, 3.47974610718975597532917081896, 4.42213437940442914855790043354, 5.06486572063518441556601930076, 5.808361587588875864770082038838, 6.53906995261716128144555437465, 7.18648498479495363320514840359, 7.79170359803542570602688835127, 8.68275427845664314273407620663, 9.397690501500734585570185208196, 9.9758831116851528291469516123, 10.561118297516096960313918319918, 11.34468574217112899267180683027, 12.43795955951072001551730977917, 13.223068347414456781798656863818, 13.91318213991436380943879023195, 14.36298593923925672892412313720, 14.83361696684164201317553970755, 15.784107493730041318878785608014, 16.63263972363865235517699108103, 17.05631188815330767233426816107, 17.66062417319307718114011263580, 18.01799930724198638435123681566

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