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Properties

Label 1-4029-4029.1007-r0-0-0
Degree $1$
Conductor $4029$
Sign $0.947 - 0.321i$
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  + (−0.845 − 0.534i)2-s + (0.428 + 0.903i)4-s + (0.774 − 0.632i)5-s + (−0.391 + 0.919i)7-s + (0.120 − 0.992i)8-s + (−0.992 + 0.120i)10-s + (0.160 − 0.987i)11-s + (−0.996 − 0.0804i)13-s + (0.822 − 0.568i)14-s + (−0.632 + 0.774i)16-s + (0.200 + 0.979i)19-s + (0.903 + 0.428i)20-s + (−0.663 + 0.748i)22-s + (0.866 + 0.5i)23-s + (0.200 − 0.979i)25-s + (0.799 + 0.600i)26-s + ⋯
L(s)  = 1  + (−0.845 − 0.534i)2-s + (0.428 + 0.903i)4-s + (0.774 − 0.632i)5-s + (−0.391 + 0.919i)7-s + (0.120 − 0.992i)8-s + (−0.992 + 0.120i)10-s + (0.160 − 0.987i)11-s + (−0.996 − 0.0804i)13-s + (0.822 − 0.568i)14-s + (−0.632 + 0.774i)16-s + (0.200 + 0.979i)19-s + (0.903 + 0.428i)20-s + (−0.663 + 0.748i)22-s + (0.866 + 0.5i)23-s + (0.200 − 0.979i)25-s + (0.799 + 0.600i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.321i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.321i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4029\)    =    \(3 \cdot 17 \cdot 79\)
Sign: $0.947 - 0.321i$
Analytic conductor: \(18.7105\)
Root analytic conductor: \(18.7105\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4029} (1007, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4029,\ (0:\ ),\ 0.947 - 0.321i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.135426004 - 0.1872944928i\)
\(L(\frac12)\) \(\approx\) \(1.135426004 - 0.1872944928i\)
\(L(1)\) \(\approx\) \(0.7783621544 - 0.1633593444i\)
\(L(1)\) \(\approx\) \(0.7783621544 - 0.1633593444i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
79 \( 1 \)
good2 \( 1 + (-0.845 - 0.534i)T \)
5 \( 1 + (0.774 - 0.632i)T \)
7 \( 1 + (-0.391 + 0.919i)T \)
11 \( 1 + (0.160 - 0.987i)T \)
13 \( 1 + (-0.996 - 0.0804i)T \)
19 \( 1 + (0.200 + 0.979i)T \)
23 \( 1 + (0.866 + 0.5i)T \)
29 \( 1 + (-0.721 - 0.692i)T \)
31 \( 1 + (0.999 + 0.0402i)T \)
37 \( 1 + (-0.316 + 0.948i)T \)
41 \( 1 + (-0.935 - 0.354i)T \)
43 \( 1 + (0.987 - 0.160i)T \)
47 \( 1 + (0.948 - 0.316i)T \)
53 \( 1 + (-0.799 - 0.600i)T \)
59 \( 1 + (-0.428 + 0.903i)T \)
61 \( 1 + (0.663 + 0.748i)T \)
67 \( 1 + (0.885 + 0.464i)T \)
71 \( 1 + (-0.992 - 0.120i)T \)
73 \( 1 + (0.0804 + 0.996i)T \)
83 \( 1 + (0.428 + 0.903i)T \)
89 \( 1 + (-0.120 - 0.992i)T \)
97 \( 1 + (0.663 + 0.748i)T \)
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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

−18.41930892865300318889085021860, −17.49259054487150558862256203487, −17.35226912342222039698251700071, −16.76618927470994226456918339957, −15.79995613125376539487607151493, −15.162745156105159204853045360256, −14.452198503565978101526742268862, −14.03312875942355373312878314396, −13.1406891302620770244969094228, −12.39705065189429445324607911199, −11.28829457252996409378882748341, −10.67449048333722058550072066161, −10.13091811980850181031265662891, −9.42825039494429437417382241271, −9.10501375735562981590428682138, −7.79748417048231786739590797103, −7.14302396257953963455413500161, −6.872615078770041042433998864556, −6.10813455357345164178949339304, −5.071193443900026137061180074749, −4.554254848710400716501743191581, −3.196640557020419210048513491506, −2.41359919965746199683739853252, −1.66332178639457008220035819533, −0.60374540594134370731318622910, 0.72206113731287376320951521766, 1.60530780948962219718214293354, 2.43713860925215409274496898848, 3.02826085195620760480239022918, 3.95790213843074144092267646646, 5.09478924061914531603461346116, 5.76751164130908289325184977487, 6.46433231954542722733396774666, 7.40177383319888386368741233518, 8.32712903263240552932869131730, 8.757637410204022401604324637100, 9.51987537493760394191664295510, 9.92407935560486025352871617305, 10.69976422827559051105955055149, 11.81575249326008629225176086486, 11.980409635419283796774681264737, 12.89416980298264336968606612017, 13.39585757786821528337521687864, 14.26247174192266784696957123392, 15.2581570254727383387059886220, 15.90820079265928984870671982103, 16.668968978942784755661179905869, 17.13257552128223569555995650597, 17.65866002087312194097387621774, 18.78919083167780220823053007479

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