Properties

Label 1-4029-4029.305-r0-0-0
Degree $1$
Conductor $4029$
Sign $0.766 + 0.641i$
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  + (0.996 − 0.0804i)2-s + (0.987 − 0.160i)4-s + (0.948 + 0.316i)5-s + (0.278 + 0.960i)7-s + (0.970 − 0.239i)8-s + (0.970 + 0.239i)10-s + (−0.200 + 0.979i)11-s + (−0.632 − 0.774i)13-s + (0.354 + 0.935i)14-s + (0.948 − 0.316i)16-s + (0.799 − 0.600i)19-s + (0.987 + 0.160i)20-s + (−0.120 + 0.992i)22-s + (−0.5 − 0.866i)23-s + (0.799 + 0.600i)25-s + (−0.692 − 0.721i)26-s + ⋯
L(s)  = 1  + (0.996 − 0.0804i)2-s + (0.987 − 0.160i)4-s + (0.948 + 0.316i)5-s + (0.278 + 0.960i)7-s + (0.970 − 0.239i)8-s + (0.970 + 0.239i)10-s + (−0.200 + 0.979i)11-s + (−0.632 − 0.774i)13-s + (0.354 + 0.935i)14-s + (0.948 − 0.316i)16-s + (0.799 − 0.600i)19-s + (0.987 + 0.160i)20-s + (−0.120 + 0.992i)22-s + (−0.5 − 0.866i)23-s + (0.799 + 0.600i)25-s + (−0.692 − 0.721i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.641i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.641i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4029\)    =    \(3 \cdot 17 \cdot 79\)
Sign: $0.766 + 0.641i$
Analytic conductor: \(18.7105\)
Root analytic conductor: \(18.7105\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4029} (305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4029,\ (0:\ ),\ 0.766 + 0.641i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.405171626 + 1.600638266i\)
\(L(\frac12)\) \(\approx\) \(4.405171626 + 1.600638266i\)
\(L(1)\) \(\approx\) \(2.421496901 + 0.3544674268i\)
\(L(1)\) \(\approx\) \(2.421496901 + 0.3544674268i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
79 \( 1 \)
good2 \( 1 + (0.996 - 0.0804i)T \)
5 \( 1 + (0.948 + 0.316i)T \)
7 \( 1 + (0.278 + 0.960i)T \)
11 \( 1 + (-0.200 + 0.979i)T \)
13 \( 1 + (-0.632 - 0.774i)T \)
19 \( 1 + (0.799 - 0.600i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.845 + 0.534i)T \)
31 \( 1 + (-0.428 + 0.903i)T \)
37 \( 1 + (-0.919 - 0.391i)T \)
41 \( 1 + (0.748 - 0.663i)T \)
43 \( 1 + (0.200 + 0.979i)T \)
47 \( 1 + (-0.919 + 0.391i)T \)
53 \( 1 + (0.692 + 0.721i)T \)
59 \( 1 + (0.987 + 0.160i)T \)
61 \( 1 + (0.120 + 0.992i)T \)
67 \( 1 + (0.568 - 0.822i)T \)
71 \( 1 + (0.970 - 0.239i)T \)
73 \( 1 + (0.632 - 0.774i)T \)
83 \( 1 + (-0.987 + 0.160i)T \)
89 \( 1 + (0.970 + 0.239i)T \)
97 \( 1 + (-0.120 - 0.992i)T \)
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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

−18.37557439894617125440461332075, −17.34031701878522255255995566707, −17.04331314604034731887268348092, −16.24521603446164135565830464662, −15.84699602226670436699870904434, −14.56925699599195181540557078120, −14.25660668061469824371498178374, −13.54046002022538086500921762878, −13.315722345973714877863465938350, −12.29059328866718634518121278132, −11.59421308704632488603917952481, −11.01125644990104139116856601401, −10.043417765762025187445994006550, −9.72426460820825096407126168085, −8.46783120476654236655459212670, −7.80398300632175409219807499013, −6.96496421494179283074279269342, −6.35064014701528594417332961986, −5.47559971938500142814265879530, −5.081272996650759783195851089464, −4.0840004591789409281194793832, −3.526411972441905542136800293124, −2.48433331089173987175470147614, −1.76374698589351270334644352168, −0.90723007505876279512497555332, 1.23391939685393757839983644906, 2.18535935166918126102723780046, 2.58567263992336477096984322554, 3.30750716518363554908542033071, 4.58034576946487175888749755048, 5.14809016662579140836271311945, 5.600280947015892229885493812721, 6.493869123040884127972326374834, 7.10063524026185939403762978277, 7.89298206986044257929106860032, 8.92401364013368972830800912675, 9.73691425634665206855568357224, 10.37154112114591444191448078972, 10.97456258677624966291585595851, 11.9844599468165003397419849542, 12.4856307435011685241002321624, 12.93741195415427871174992440499, 13.912940147671640820376460570343, 14.44584955073219947286739940844, 14.952265078435895432408047242356, 15.66752219299300186583745200684, 16.24899922116675919383807166073, 17.31663427313698894039797396932, 17.92823951193296549171051096465, 18.2987440712008636556659126930

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