Properties

Label 1-4029-4029.689-r0-0-0
Degree $1$
Conductor $4029$
Sign $0.455 - 0.890i$
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  + (0.935 − 0.354i)2-s + (0.748 − 0.663i)4-s + (−0.616 + 0.787i)5-s + (0.297 − 0.954i)7-s + (0.464 − 0.885i)8-s + (−0.297 + 0.954i)10-s + (0.616 + 0.787i)11-s + (0.748 − 0.663i)13-s + (−0.0603 − 0.998i)14-s + (0.120 − 0.992i)16-s + (−0.239 + 0.970i)19-s + (0.0603 + 0.998i)20-s + (0.855 + 0.517i)22-s + (−0.707 + 0.707i)23-s + (−0.239 − 0.970i)25-s + (0.464 − 0.885i)26-s + ⋯
L(s)  = 1  + (0.935 − 0.354i)2-s + (0.748 − 0.663i)4-s + (−0.616 + 0.787i)5-s + (0.297 − 0.954i)7-s + (0.464 − 0.885i)8-s + (−0.297 + 0.954i)10-s + (0.616 + 0.787i)11-s + (0.748 − 0.663i)13-s + (−0.0603 − 0.998i)14-s + (0.120 − 0.992i)16-s + (−0.239 + 0.970i)19-s + (0.0603 + 0.998i)20-s + (0.855 + 0.517i)22-s + (−0.707 + 0.707i)23-s + (−0.239 − 0.970i)25-s + (0.464 − 0.885i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.455 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.455 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4029\)    =    \(3 \cdot 17 \cdot 79\)
Sign: $0.455 - 0.890i$
Analytic conductor: \(18.7105\)
Root analytic conductor: \(18.7105\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4029} (689, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4029,\ (0:\ ),\ 0.455 - 0.890i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.826595796 - 1.729764157i\)
\(L(\frac12)\) \(\approx\) \(2.826595796 - 1.729764157i\)
\(L(1)\) \(\approx\) \(1.806601392 - 0.5241545078i\)
\(L(1)\) \(\approx\) \(1.806601392 - 0.5241545078i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
79 \( 1 \)
good2 \( 1 + (0.935 - 0.354i)T \)
5 \( 1 + (-0.616 + 0.787i)T \)
7 \( 1 + (0.297 - 0.954i)T \)
11 \( 1 + (0.616 + 0.787i)T \)
13 \( 1 + (0.748 - 0.663i)T \)
19 \( 1 + (-0.239 + 0.970i)T \)
23 \( 1 + (-0.707 + 0.707i)T \)
29 \( 1 + (0.180 - 0.983i)T \)
31 \( 1 + (0.410 - 0.911i)T \)
37 \( 1 + (0.517 + 0.855i)T \)
41 \( 1 + (-0.787 - 0.616i)T \)
43 \( 1 + (0.992 - 0.120i)T \)
47 \( 1 + (0.970 - 0.239i)T \)
53 \( 1 + (0.464 - 0.885i)T \)
59 \( 1 + (0.663 - 0.748i)T \)
61 \( 1 + (-0.855 + 0.517i)T \)
67 \( 1 + (-0.354 + 0.935i)T \)
71 \( 1 + (0.297 + 0.954i)T \)
73 \( 1 + (0.0603 + 0.998i)T \)
83 \( 1 + (-0.663 - 0.748i)T \)
89 \( 1 + (0.885 - 0.464i)T \)
97 \( 1 + (0.517 + 0.855i)T \)
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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

−18.566909317991079594746931424733, −17.85036516413808915695338443102, −16.886278190337749413271467088, −16.38351151208240225263612335279, −15.81190792659573763942836591856, −15.2939234387113126688661579565, −14.44129037388376443059005756229, −13.88461027210479520421304559964, −13.147665461449734713993163453782, −12.27076825999643042517413610896, −12.04253716002534525068971893955, −11.215268227619516574234155604, −10.75019214655097313257541979727, −9.071467020342242372272909874420, −8.85476546393511630325276515599, −8.17893674660856469733766372262, −7.32499222461414577497653990613, −6.376070737023683801869328745932, −5.93233998059379255655726140796, −4.99328992751132014249477365601, −4.47004420428044694230368603136, −3.71100006227141847531872601760, −2.922181012427021763851658876930, −1.978064008300924973622476887142, −1.04042618683958740787458884459, 0.74637260832329242203780762502, 1.69460543973961860300205145285, 2.53101622885869768620211582800, 3.59177278253857715573917291619, 3.94784707606796287993418541335, 4.50922984252390866635521769363, 5.69805057007652845788118046591, 6.28183545708050040736986978040, 7.09954722944319129307423041855, 7.63791068017834517845512289569, 8.36618422074754736280938148568, 9.91634960904410261227625680180, 10.10028364439365423213728658440, 10.91825060199684462634664651647, 11.60935406979919743208566581414, 12.01640298306034569844003390484, 12.97392973186502909204346413090, 13.64200038820789129459219846728, 14.23758808329157197360471017559, 14.85471160153147854089047619846, 15.44592773973746862552437547731, 16.03449419807584847523518369036, 16.99876080878945732344979291937, 17.632700245928000381813359890408, 18.580957769743357190132777654628

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