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Properties

Label 1-3381-3381.416-r0-0-0
Degree $1$
Conductor $3381$
Sign $0.335 + 0.941i$
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  + (−0.855 − 0.517i)2-s + (0.464 + 0.885i)4-s + (0.644 + 0.764i)5-s + (0.0611 − 0.998i)8-s + (−0.155 − 0.987i)10-s + (−0.938 − 0.346i)11-s + (−0.557 + 0.830i)13-s + (−0.568 + 0.822i)16-s + (0.534 + 0.844i)17-s + (0.327 + 0.945i)19-s + (−0.377 + 0.925i)20-s + (0.623 + 0.781i)22-s + (−0.169 + 0.985i)25-s + (0.906 − 0.421i)26-s + (0.999 − 0.0407i)29-s + ⋯
L(s)  = 1  + (−0.855 − 0.517i)2-s + (0.464 + 0.885i)4-s + (0.644 + 0.764i)5-s + (0.0611 − 0.998i)8-s + (−0.155 − 0.987i)10-s + (−0.938 − 0.346i)11-s + (−0.557 + 0.830i)13-s + (−0.568 + 0.822i)16-s + (0.534 + 0.844i)17-s + (0.327 + 0.945i)19-s + (−0.377 + 0.925i)20-s + (0.623 + 0.781i)22-s + (−0.169 + 0.985i)25-s + (0.906 − 0.421i)26-s + (0.999 − 0.0407i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.335 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.335 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.335 + 0.941i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (416, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ 0.335 + 0.941i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9040794662 + 0.6376406555i\)
\(L(\frac12)\) \(\approx\) \(0.9040794662 + 0.6376406555i\)
\(L(1)\) \(\approx\) \(0.7889268709 + 0.08321153136i\)
\(L(1)\) \(\approx\) \(0.7889268709 + 0.08321153136i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.855 - 0.517i)T \)
5 \( 1 + (0.644 + 0.764i)T \)
11 \( 1 + (-0.938 - 0.346i)T \)
13 \( 1 + (-0.557 + 0.830i)T \)
17 \( 1 + (0.534 + 0.844i)T \)
19 \( 1 + (0.327 + 0.945i)T \)
29 \( 1 + (0.999 - 0.0407i)T \)
31 \( 1 + (-0.235 - 0.971i)T \)
37 \( 1 + (0.810 + 0.585i)T \)
41 \( 1 + (0.339 - 0.940i)T \)
43 \( 1 + (-0.0611 - 0.998i)T \)
47 \( 1 + (0.826 - 0.563i)T \)
53 \( 1 + (0.973 + 0.229i)T \)
59 \( 1 + (0.155 + 0.987i)T \)
61 \( 1 + (0.0882 + 0.996i)T \)
67 \( 1 + (0.580 - 0.814i)T \)
71 \( 1 + (0.452 - 0.891i)T \)
73 \( 1 + (0.966 - 0.255i)T \)
79 \( 1 + (-0.786 - 0.618i)T \)
83 \( 1 + (-0.523 + 0.852i)T \)
89 \( 1 + (0.209 + 0.977i)T \)
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

−18.299434974739378965764393166921, −17.97905287054937844643124035473, −17.40546367691092544080577913619, −16.68120526881857274095840171701, −15.86412489343455775699892038309, −15.65270443851083358400118277822, −14.53174202273685823821122245873, −14.02860476379353226818072428458, −13.05343496705447919898170837877, −12.545747552804911018582656431563, −11.55217464002904201889225911260, −10.7508010815237259133323117867, −9.87105881631284607944118182356, −9.67953207574662762209964968698, −8.73788764929163484957989777926, −8.055985131419757534438580456097, −7.418577286908339918100100746899, −6.644991973040385450128120591381, −5.65201677073486796499828802694, −5.152723072051304316091666841671, −4.60608337485834631377593360202, −2.85335813283377596773427118785, −2.4389589886989531892807106411, −1.23104198629817628119804261107, −0.522069952930735957694496037524, 0.98652744901819315014752022868, 2.06621827494547967142529119032, 2.48955032407906347233074763713, 3.423612820699988723936039541606, 4.166096952335979007645804795135, 5.47016749501927882963979991793, 6.13009490621825067525842181960, 7.015938499620146059423028990768, 7.6468051281738187149246526550, 8.360625253687691144855615349120, 9.21814352905532986156783099843, 10.004352532948665195575746977238, 10.36913329704201469405754366542, 11.0343597242680519596623606710, 11.92183986554918782252061492274, 12.45554930847725214611644085110, 13.44294009192637628318604768375, 13.94361089925771094399425329824, 14.916258208225180781459105779543, 15.52056221341407409772089484457, 16.68063541088757320516508333112, 16.77423765969200364848263053770, 17.77458440393753668013648113971, 18.36339661667735786462652191363, 18.88710784427696782795763553355

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