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Properties

Label 1-3381-3381.1235-r0-0-0
Degree $1$
Conductor $3381$
Sign $-0.394 - 0.919i$
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  + (−0.155 + 0.987i)2-s + (−0.951 − 0.307i)4-s + (0.534 − 0.844i)5-s + (0.452 − 0.891i)8-s + (0.751 + 0.659i)10-s + (0.568 + 0.822i)11-s + (−0.339 + 0.940i)13-s + (0.810 + 0.585i)16-s + (0.209 − 0.977i)17-s + (−0.235 + 0.971i)19-s + (−0.768 + 0.639i)20-s + (−0.900 + 0.433i)22-s + (−0.427 − 0.903i)25-s + (−0.876 − 0.482i)26-s + (−0.742 − 0.670i)29-s + ⋯
L(s)  = 1  + (−0.155 + 0.987i)2-s + (−0.951 − 0.307i)4-s + (0.534 − 0.844i)5-s + (0.452 − 0.891i)8-s + (0.751 + 0.659i)10-s + (0.568 + 0.822i)11-s + (−0.339 + 0.940i)13-s + (0.810 + 0.585i)16-s + (0.209 − 0.977i)17-s + (−0.235 + 0.971i)19-s + (−0.768 + 0.639i)20-s + (−0.900 + 0.433i)22-s + (−0.427 − 0.903i)25-s + (−0.876 − 0.482i)26-s + (−0.742 − 0.670i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.394 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.394 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $-0.394 - 0.919i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (1235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ -0.394 - 0.919i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1004121783 - 0.1523307387i\)
\(L(\frac12)\) \(\approx\) \(0.1004121783 - 0.1523307387i\)
\(L(1)\) \(\approx\) \(0.7602300795 + 0.2610724428i\)
\(L(1)\) \(\approx\) \(0.7602300795 + 0.2610724428i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.155 + 0.987i)T \)
5 \( 1 + (0.534 - 0.844i)T \)
11 \( 1 + (0.568 + 0.822i)T \)
13 \( 1 + (-0.339 + 0.940i)T \)
17 \( 1 + (0.209 - 0.977i)T \)
19 \( 1 + (-0.235 + 0.971i)T \)
29 \( 1 + (-0.742 - 0.670i)T \)
31 \( 1 + (-0.580 + 0.814i)T \)
37 \( 1 + (-0.966 - 0.255i)T \)
41 \( 1 + (-0.999 - 0.0407i)T \)
43 \( 1 + (-0.452 - 0.891i)T \)
47 \( 1 + (0.955 + 0.294i)T \)
53 \( 1 + (-0.999 - 0.0271i)T \)
59 \( 1 + (-0.751 - 0.659i)T \)
61 \( 1 + (-0.855 - 0.517i)T \)
67 \( 1 + (-0.786 + 0.618i)T \)
71 \( 1 + (-0.557 - 0.830i)T \)
73 \( 1 + (0.833 - 0.552i)T \)
79 \( 1 + (-0.888 + 0.458i)T \)
83 \( 1 + (0.882 - 0.470i)T \)
89 \( 1 + (-0.923 + 0.384i)T \)
97 \( 1 + (0.654 - 0.755i)T \)
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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

−19.00640262566009854782878071216, −18.51943172549606724759837412773, −17.74771213952189026121544155784, −17.17308726898504387964444856218, −16.6466145264310438749451706303, −15.27523086814287690208804157168, −14.8287235189441465780164104914, −14.01049455810125587220909058392, −13.396348958351143885042705828333, −12.799814348297309678016519534838, −11.96689502193499126656312880455, −11.10276250398841709662151064780, −10.742260611533204353496692551501, −10.05701731110836031186540964961, −9.28983069955636616571218593529, −8.629085627833562352513441285032, −7.79740864746861596976394843264, −6.94551936675356429627045749386, −5.96003897575230728119721611839, −5.39114664630241314050308086939, −4.332457611332010368065435160058, −3.363567853519470850973210048700, −3.02272589393501144759611889963, −2.017157307076311226202572270038, −1.26661743942872895064655980974, 0.05570242607234492867938944397, 1.453417105558773011198578679903, 1.95312281500509427867297992799, 3.52396586640495748793338065277, 4.3729856518345484258317777294, 4.9507416122995418519619438407, 5.646810976880336401949707876, 6.485604330759942203440585237467, 7.14343114009656264593594166774, 7.85959014832679024284741241745, 8.79881588609863015112770615315, 9.30846247280834639140750536694, 9.79161903264111080976221448708, 10.619440616710240481443601804356, 12.01162339125647892622995524338, 12.278191445894273488847329988723, 13.226298861169729678506442991462, 14.00229049484470939167116797936, 14.32141423772500266395159192457, 15.23487943721577458327173447094, 15.9560514726737801303975175623, 16.69386636162114774983175965151, 17.03367973289312211957029378367, 17.67887415370954649673637209398, 18.51481220273505749406770822523

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