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Properties

Label 1-3381-3381.209-r0-0-0
Degree $1$
Conductor $3381$
Sign $-0.886 + 0.462i$
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  + (0.768 + 0.639i)2-s + (0.182 + 0.983i)4-s + (−0.714 + 0.699i)5-s + (−0.488 + 0.872i)8-s + (−0.996 + 0.0815i)10-s + (−0.0203 − 0.999i)11-s + (0.301 + 0.953i)13-s + (−0.933 + 0.359i)16-s + (0.182 − 0.983i)17-s + (0.654 − 0.755i)19-s + (−0.818 − 0.574i)20-s + (0.623 − 0.781i)22-s + (0.0203 − 0.999i)25-s + (−0.377 + 0.925i)26-s + (−0.182 + 0.983i)29-s + ⋯
L(s)  = 1  + (0.768 + 0.639i)2-s + (0.182 + 0.983i)4-s + (−0.714 + 0.699i)5-s + (−0.488 + 0.872i)8-s + (−0.996 + 0.0815i)10-s + (−0.0203 − 0.999i)11-s + (0.301 + 0.953i)13-s + (−0.933 + 0.359i)16-s + (0.182 − 0.983i)17-s + (0.654 − 0.755i)19-s + (−0.818 − 0.574i)20-s + (0.623 − 0.781i)22-s + (0.0203 − 0.999i)25-s + (−0.377 + 0.925i)26-s + (−0.182 + 0.983i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.886 + 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.886 + 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $-0.886 + 0.462i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ -0.886 + 0.462i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4799543211 + 1.956300507i\)
\(L(\frac12)\) \(\approx\) \(0.4799543211 + 1.956300507i\)
\(L(1)\) \(\approx\) \(1.118147374 + 0.8258069989i\)
\(L(1)\) \(\approx\) \(1.118147374 + 0.8258069989i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.768 + 0.639i)T \)
5 \( 1 + (-0.714 + 0.699i)T \)
11 \( 1 + (-0.0203 - 0.999i)T \)
13 \( 1 + (0.301 + 0.953i)T \)
17 \( 1 + (0.182 - 0.983i)T \)
19 \( 1 + (0.654 - 0.755i)T \)
29 \( 1 + (-0.182 + 0.983i)T \)
31 \( 1 + (0.959 + 0.281i)T \)
37 \( 1 + (0.947 + 0.320i)T \)
41 \( 1 + (-0.714 + 0.699i)T \)
43 \( 1 + (0.488 + 0.872i)T \)
47 \( 1 + (-0.900 + 0.433i)T \)
53 \( 1 + (0.862 + 0.505i)T \)
59 \( 1 + (0.996 - 0.0815i)T \)
61 \( 1 + (0.377 + 0.925i)T \)
67 \( 1 + (0.415 + 0.909i)T \)
71 \( 1 + (-0.970 - 0.242i)T \)
73 \( 1 + (-0.917 + 0.396i)T \)
79 \( 1 + (-0.142 + 0.989i)T \)
83 \( 1 + (-0.992 + 0.122i)T \)
89 \( 1 + (0.986 + 0.162i)T \)
97 \( 1 + (-0.841 - 0.540i)T \)
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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

−18.82037329163331332636503980030, −17.8862597909245961823049118389, −17.18219327834110718367775881334, −16.2479500452093855204324163743, −15.53329326854793233987243790441, −15.08340780476328698958456652787, −14.43329151493283267466028617143, −13.29319464685863893600895720003, −13.030250971520489755566131985975, −12.09707278440530256575125538213, −11.90563955045670741197863236107, −10.909348960922337542978585776742, −10.12816271500968806907084577754, −9.65694892439447417298569013334, −8.55975984831259886994108003956, −7.89120960955152742960154645661, −7.09093335378054580091890705872, −6.011635952658720348133786568, −5.43218253057387802036465838725, −4.61740646787877487536990541039, −3.93130821215698748957315591608, −3.36154362175360462663668807442, −2.25367045524965573103105059038, −1.444265503874776108892900253015, −0.502093728776214906362492100470, 1.09531403262630849856843112762, 2.72774595344136575387755773938, 2.97810323054345502099630438966, 3.93121448391217411214003240287, 4.60847621769388339828370770132, 5.43711141839239470172238584482, 6.34865208256290287111639869485, 6.89823265904736699294465021568, 7.51767470018132025220338120579, 8.35196477046604370427977715950, 8.95590136739877978983509041141, 9.97476437140636638361141161235, 11.20357090904920208486383911130, 11.42827638641748828622286192435, 12.01022179589643629425089959161, 13.10960169057826893449148609906, 13.68641199039340926003858518835, 14.326707412140401446385464916, 14.834185757922688221157179016567, 15.796165395406047511141215531458, 16.16723183453975935314227703322, 16.66364061813752236027627049879, 17.80377885058828813751806016136, 18.3183284153101586642370224915, 19.06325094285377135385411429883

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