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Properties

Label 1-3381-3381.1031-r0-0-0
Degree $1$
Conductor $3381$
Sign $0.949 - 0.312i$
Analytic cond. $15.7012$
Root an. cond. $15.7012$
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  + (0.966 − 0.255i)2-s + (0.869 − 0.494i)4-s + (0.665 + 0.746i)5-s + (0.714 − 0.699i)8-s + (0.833 + 0.552i)10-s + (−0.802 + 0.596i)11-s + (0.986 − 0.162i)13-s + (0.511 − 0.859i)16-s + (−0.00679 − 0.999i)17-s + (0.786 − 0.618i)19-s + (0.947 + 0.320i)20-s + (−0.623 + 0.781i)22-s + (−0.115 + 0.993i)25-s + (0.912 − 0.409i)26-s + (0.862 − 0.505i)29-s + ⋯
L(s)  = 1  + (0.966 − 0.255i)2-s + (0.869 − 0.494i)4-s + (0.665 + 0.746i)5-s + (0.714 − 0.699i)8-s + (0.833 + 0.552i)10-s + (−0.802 + 0.596i)11-s + (0.986 − 0.162i)13-s + (0.511 − 0.859i)16-s + (−0.00679 − 0.999i)17-s + (0.786 − 0.618i)19-s + (0.947 + 0.320i)20-s + (−0.623 + 0.781i)22-s + (−0.115 + 0.993i)25-s + (0.912 − 0.409i)26-s + (0.862 − 0.505i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.949 - 0.312i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (1031, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ 0.949 - 0.312i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.146296442 - 0.6648169814i\)
\(L(\frac12)\) \(\approx\) \(4.146296442 - 0.6648169814i\)
\(L(1)\) \(\approx\) \(2.262723698 - 0.2336136090i\)
\(L(1)\) \(\approx\) \(2.262723698 - 0.2336136090i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.966 - 0.255i)T \)
5 \( 1 + (0.665 + 0.746i)T \)
11 \( 1 + (-0.802 + 0.596i)T \)
13 \( 1 + (0.986 - 0.162i)T \)
17 \( 1 + (-0.00679 - 0.999i)T \)
19 \( 1 + (0.786 - 0.618i)T \)
29 \( 1 + (0.862 - 0.505i)T \)
31 \( 1 + (-0.888 + 0.458i)T \)
37 \( 1 + (0.694 + 0.719i)T \)
41 \( 1 + (0.979 - 0.202i)T \)
43 \( 1 + (0.714 + 0.699i)T \)
47 \( 1 + (-0.826 - 0.563i)T \)
53 \( 1 + (-0.612 - 0.790i)T \)
59 \( 1 + (0.833 + 0.552i)T \)
61 \( 1 + (-0.810 + 0.585i)T \)
67 \( 1 + (0.327 + 0.945i)T \)
71 \( 1 + (-0.182 - 0.983i)T \)
73 \( 1 + (-0.675 - 0.737i)T \)
79 \( 1 + (-0.235 - 0.971i)T \)
83 \( 1 + (-0.768 + 0.639i)T \)
89 \( 1 + (0.601 + 0.798i)T \)
97 \( 1 + (-0.415 - 0.909i)T \)
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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

−18.82181164356224581922157320349, −18.05338199638952706359918939293, −17.317392036062128804072151932974, −16.54549628068078894752211994764, −16.02524631837206756840154315122, −15.58555835034045056115230957861, −14.317843057405389111765580523077, −14.14905399709099540967984886427, −13.094765386837379921580375299586, −12.92600067665478567731424222019, −12.13994209647748780212013429521, −11.136710151494934877014184796098, −10.690463347574897376074413491518, −9.73639497337224555794087826580, −8.75909924116138885099270872714, −8.15612285719324636651593869170, −7.45800816355392662607640128173, −6.267584114603827270743841389924, −5.87775280448717195798996309855, −5.28847887275270080444639830207, −4.38118214438468436582868752515, −3.663168562714258162480252254986, −2.80480017983154771549017458100, −1.87006775901888009453651959805, −1.08546607538068689608950278440, 0.98790803035942942177171123876, 1.97076868250961152038703028426, 2.808298748732454854322980908172, 3.19045211347280254541215749145, 4.33333743191760932924527214307, 5.09010518315450988904788118658, 5.7498010944056600488480071118, 6.483802830979245451212606712009, 7.192587833962791603705551429356, 7.82051036789095743405061508391, 9.12137696116462283975312842262, 9.88028620941605214783708423412, 10.460145380092829646397305663583, 11.20844885864042526320714810689, 11.69496787587317216228350106053, 12.7444890260817797993530919772, 13.33464509679662790504599298708, 13.80133354386357531538884034248, 14.53450930549941012047338398264, 15.187876491049961429407434688922, 15.940157108590021800671974143351, 16.3467485165872445404675157446, 17.732406908166706150735415086839, 18.02960953234102008957635860104, 18.76252310427123139959411304371

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