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Properties

Label 1-4029-4029.1913-r0-0-0
Degree $1$
Conductor $4029$
Sign $0.187 + 0.982i$
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  + (−0.464 + 0.885i)2-s + (−0.568 − 0.822i)4-s + (0.410 − 0.911i)5-s + (−0.616 − 0.787i)7-s + (0.992 − 0.120i)8-s + (0.616 + 0.787i)10-s + (−0.410 − 0.911i)11-s + (−0.568 − 0.822i)13-s + (0.983 − 0.180i)14-s + (−0.354 + 0.935i)16-s + (−0.663 + 0.748i)19-s + (−0.983 + 0.180i)20-s + (0.998 + 0.0603i)22-s + (−0.707 + 0.707i)23-s + (−0.663 − 0.748i)25-s + (0.992 − 0.120i)26-s + ⋯
L(s)  = 1  + (−0.464 + 0.885i)2-s + (−0.568 − 0.822i)4-s + (0.410 − 0.911i)5-s + (−0.616 − 0.787i)7-s + (0.992 − 0.120i)8-s + (0.616 + 0.787i)10-s + (−0.410 − 0.911i)11-s + (−0.568 − 0.822i)13-s + (0.983 − 0.180i)14-s + (−0.354 + 0.935i)16-s + (−0.663 + 0.748i)19-s + (−0.983 + 0.180i)20-s + (0.998 + 0.0603i)22-s + (−0.707 + 0.707i)23-s + (−0.663 − 0.748i)25-s + (0.992 − 0.120i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4029\)    =    \(3 \cdot 17 \cdot 79\)
Sign: $0.187 + 0.982i$
Analytic conductor: \(18.7105\)
Root analytic conductor: \(18.7105\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4029} (1913, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4029,\ (0:\ ),\ 0.187 + 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4706788368 + 0.3894050874i\)
\(L(\frac12)\) \(\approx\) \(0.4706788368 + 0.3894050874i\)
\(L(1)\) \(\approx\) \(0.6763186818 + 0.06887466470i\)
\(L(1)\) \(\approx\) \(0.6763186818 + 0.06887466470i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
79 \( 1 \)
good2 \( 1 + (-0.464 + 0.885i)T \)
5 \( 1 + (0.410 - 0.911i)T \)
7 \( 1 + (-0.616 - 0.787i)T \)
11 \( 1 + (-0.410 - 0.911i)T \)
13 \( 1 + (-0.568 - 0.822i)T \)
19 \( 1 + (-0.663 + 0.748i)T \)
23 \( 1 + (-0.707 + 0.707i)T \)
29 \( 1 + (0.855 + 0.517i)T \)
31 \( 1 + (0.297 + 0.954i)T \)
37 \( 1 + (0.0603 + 0.998i)T \)
41 \( 1 + (0.911 + 0.410i)T \)
43 \( 1 + (-0.935 + 0.354i)T \)
47 \( 1 + (0.748 - 0.663i)T \)
53 \( 1 + (0.992 - 0.120i)T \)
59 \( 1 + (0.822 + 0.568i)T \)
61 \( 1 + (-0.998 + 0.0603i)T \)
67 \( 1 + (0.885 - 0.464i)T \)
71 \( 1 + (-0.616 + 0.787i)T \)
73 \( 1 + (-0.983 + 0.180i)T \)
83 \( 1 + (-0.822 + 0.568i)T \)
89 \( 1 + (0.120 - 0.992i)T \)
97 \( 1 + (0.0603 + 0.998i)T \)
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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

−18.4182193518820105200995930286, −17.77830721059429355202549367734, −17.291275569699212424295817377752, −16.399111587767921030539939164244, −15.597202419587325252609329197849, −14.88358256626154471826525113238, −14.15071589087967945688740665551, −13.38826734087475896686755288655, −12.69421716862123234191388467229, −12.07399614512011681335691896199, −11.473816359602826804622875499854, −10.57513174774444135371183922461, −10.06959965681088040287910918603, −9.45147593321482100770627405725, −8.889973321758088254996974109118, −7.91736998380931720259050462649, −7.12351514433205543736776529239, −6.5153924608856293740876771858, −5.60685566005021699403670188285, −4.54140226564959998842716617969, −3.96130966647719839062403393772, −2.73360433332157888060334490191, −2.43732537777466472832416981928, −1.88002519376396900213685493266, −0.26242797847772882925600757740, 0.783638818996918154690511866387, 1.45540938439293369457132889814, 2.7475285768635493099643824843, 3.78097511742671719923623806611, 4.5952594610419722202509040960, 5.38094371931954974886853376747, 5.936785412273228850555394178861, 6.65483620347659171274101267650, 7.53346904357031312640747575292, 8.255428917040790875123976452740, 8.65720604151151510536692203972, 9.65299822522787146252484717949, 10.20601716973799960983343097741, 10.58707658495069430004056488923, 11.84451084362440972727898006815, 12.74514739505830805603476664116, 13.30175681652108940238333477353, 13.84905126723157500289521878096, 14.5177723475529244964309901759, 15.48923421615497076050772339103, 16.08802376562961050348973222341, 16.58167121028697987689854975056, 17.13287494091325360979822825842, 17.74896848282577844121705020939, 18.43713494521475025090814865756

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