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Properties

Label 1-4029-4029.1844-r0-0-0
Degree $1$
Conductor $4029$
Sign $0.997 - 0.0700i$
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  + (−0.822 + 0.568i)2-s + (0.354 − 0.935i)4-s + (0.998 + 0.0603i)5-s + (0.855 − 0.517i)7-s + (0.239 + 0.970i)8-s + (−0.855 + 0.517i)10-s + (−0.998 + 0.0603i)11-s + (0.354 − 0.935i)13-s + (−0.410 + 0.911i)14-s + (−0.748 − 0.663i)16-s + (0.992 − 0.120i)19-s + (0.410 − 0.911i)20-s + (0.787 − 0.616i)22-s + (0.707 − 0.707i)23-s + (0.992 + 0.120i)25-s + (0.239 + 0.970i)26-s + ⋯
L(s)  = 1  + (−0.822 + 0.568i)2-s + (0.354 − 0.935i)4-s + (0.998 + 0.0603i)5-s + (0.855 − 0.517i)7-s + (0.239 + 0.970i)8-s + (−0.855 + 0.517i)10-s + (−0.998 + 0.0603i)11-s + (0.354 − 0.935i)13-s + (−0.410 + 0.911i)14-s + (−0.748 − 0.663i)16-s + (0.992 − 0.120i)19-s + (0.410 − 0.911i)20-s + (0.787 − 0.616i)22-s + (0.707 − 0.707i)23-s + (0.992 + 0.120i)25-s + (0.239 + 0.970i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0700i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0700i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4029\)    =    \(3 \cdot 17 \cdot 79\)
Sign: $0.997 - 0.0700i$
Analytic conductor: \(18.7105\)
Root analytic conductor: \(18.7105\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4029} (1844, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4029,\ (0:\ ),\ 0.997 - 0.0700i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.758247359 - 0.06168208564i\)
\(L(\frac12)\) \(\approx\) \(1.758247359 - 0.06168208564i\)
\(L(1)\) \(\approx\) \(1.031087205 + 0.08251060951i\)
\(L(1)\) \(\approx\) \(1.031087205 + 0.08251060951i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
79 \( 1 \)
good2 \( 1 + (-0.822 + 0.568i)T \)
5 \( 1 + (0.998 + 0.0603i)T \)
7 \( 1 + (0.855 - 0.517i)T \)
11 \( 1 + (-0.998 + 0.0603i)T \)
13 \( 1 + (0.354 - 0.935i)T \)
19 \( 1 + (0.992 - 0.120i)T \)
23 \( 1 + (0.707 - 0.707i)T \)
29 \( 1 + (0.954 + 0.297i)T \)
31 \( 1 + (0.180 - 0.983i)T \)
37 \( 1 + (-0.616 + 0.787i)T \)
41 \( 1 + (-0.0603 + 0.998i)T \)
43 \( 1 + (0.663 + 0.748i)T \)
47 \( 1 + (-0.120 + 0.992i)T \)
53 \( 1 + (0.239 + 0.970i)T \)
59 \( 1 + (0.935 - 0.354i)T \)
61 \( 1 + (-0.787 - 0.616i)T \)
67 \( 1 + (0.568 - 0.822i)T \)
71 \( 1 + (0.855 + 0.517i)T \)
73 \( 1 + (0.410 - 0.911i)T \)
83 \( 1 + (-0.935 - 0.354i)T \)
89 \( 1 + (-0.970 - 0.239i)T \)
97 \( 1 + (-0.616 + 0.787i)T \)
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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

−18.36929710444834656235426406758, −17.8605748694227177699378732873, −17.4370189231330579949536693042, −16.59092772133897318578871417596, −15.89478108955767026134450079893, −15.30582325274156888389582650402, −14.048549568893229446209516311460, −13.817535485885448362455328968079, −12.87313321680140159423652966441, −12.176136427345627079070177915948, −11.49545861474536641569248711439, −10.81893624169902471662320101137, −10.18620063064745335694544765580, −9.49153618198141584892404839754, −8.73381781869673745113331906306, −8.383263194143261675174477147305, −7.30033858383993539131520037274, −6.810230085985893541982749226072, −5.57370789970989375917656994887, −5.19638912163319804228294017362, −4.10601637712684868906901001961, −3.06497619588505226063494151620, −2.34016598007354250176065840714, −1.723494075966061083937174881762, −0.9510271950652532120383509266, 0.82138030261518053732405740408, 1.36078783858028361037407776663, 2.44385166706155616126213932799, 3.03657979165821836995925515931, 4.6720270490406388383861100275, 5.09949499155040983843120014111, 5.834563550005135383525450991075, 6.56332801983811629477977944801, 7.41020572901589137215557857628, 8.03749691093660094944702956495, 8.55920577112842566223067634016, 9.5871871666830509530249949440, 10.03354096916617586679377338775, 10.870362026478713227156674835949, 11.0522833871127400237197433354, 12.322984490465453128892501457059, 13.27029563379478100092305498566, 13.78207822787949971240289304034, 14.459021924279299884372063123904, 15.150695780638498929273564302897, 15.80822064732315073769207031241, 16.569924781918054930459157579229, 17.291367845913924388578666317669, 17.73324720612204190927160310839, 18.31324476143447868486255925984

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