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Properties

Label 1-3381-3381.3299-r0-0-0
Degree $1$
Conductor $3381$
Sign $-0.338 + 0.940i$
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  + (0.568 + 0.822i)2-s + (−0.352 + 0.935i)4-s + (−0.942 + 0.333i)5-s + (−0.970 + 0.242i)8-s + (−0.810 − 0.585i)10-s + (0.155 − 0.987i)11-s + (−0.714 − 0.699i)13-s + (−0.751 − 0.659i)16-s + (−0.634 + 0.773i)17-s + (−0.235 − 0.971i)19-s + (0.0203 − 0.999i)20-s + (0.900 − 0.433i)22-s + (0.777 − 0.628i)25-s + (0.169 − 0.985i)26-s + (−0.986 − 0.162i)29-s + ⋯
L(s)  = 1  + (0.568 + 0.822i)2-s + (−0.352 + 0.935i)4-s + (−0.942 + 0.333i)5-s + (−0.970 + 0.242i)8-s + (−0.810 − 0.585i)10-s + (0.155 − 0.987i)11-s + (−0.714 − 0.699i)13-s + (−0.751 − 0.659i)16-s + (−0.634 + 0.773i)17-s + (−0.235 − 0.971i)19-s + (0.0203 − 0.999i)20-s + (0.900 − 0.433i)22-s + (0.777 − 0.628i)25-s + (0.169 − 0.985i)26-s + (−0.986 − 0.162i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.338 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.338 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $-0.338 + 0.940i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (3299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ -0.338 + 0.940i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6766956850 + 0.9628358810i\)
\(L(\frac12)\) \(\approx\) \(0.6766956850 + 0.9628358810i\)
\(L(1)\) \(\approx\) \(0.8580263501 + 0.4832098662i\)
\(L(1)\) \(\approx\) \(0.8580263501 + 0.4832098662i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.568 + 0.822i)T \)
5 \( 1 + (-0.942 + 0.333i)T \)
11 \( 1 + (0.155 - 0.987i)T \)
13 \( 1 + (-0.714 - 0.699i)T \)
17 \( 1 + (-0.634 + 0.773i)T \)
19 \( 1 + (-0.235 - 0.971i)T \)
29 \( 1 + (-0.986 - 0.162i)T \)
31 \( 1 + (0.580 + 0.814i)T \)
37 \( 1 + (0.802 + 0.596i)T \)
41 \( 1 + (-0.182 + 0.983i)T \)
43 \( 1 + (-0.970 - 0.242i)T \)
47 \( 1 + (0.733 - 0.680i)T \)
53 \( 1 + (0.601 - 0.798i)T \)
59 \( 1 + (-0.810 - 0.585i)T \)
61 \( 1 + (-0.938 - 0.346i)T \)
67 \( 1 + (0.786 + 0.618i)T \)
71 \( 1 + (0.301 + 0.953i)T \)
73 \( 1 + (0.511 + 0.859i)T \)
79 \( 1 + (0.888 + 0.458i)T \)
83 \( 1 + (-0.591 - 0.806i)T \)
89 \( 1 + (0.665 + 0.746i)T \)
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

−18.615926534745888287357197181708, −18.34212163506133690038167823165, −17.1441840421155828483605230031, −16.581103297318921417455241970117, −15.538046384458222157553405973611, −15.11964943101844100527595740725, −14.4339710839534377165392349979, −13.67856612609723318307670438894, −12.83696328292633500132254546454, −12.19724923135247425596021055509, −11.83081638376572079443173687658, −11.07479677436302545733226311119, −10.31427218062557678087960650852, −9.36448067501652246336927538774, −9.09291370647613897016884052422, −7.84923640428490607847103688139, −7.228263705459478586915408058615, −6.34640322316695032165652402862, −5.360775367572899251659981439711, −4.46893586665400906296548539843, −4.26581841974419691800372026334, −3.310509910915272723365389075430, −2.3138005289479947134621086784, −1.674033394481888005134824146029, −0.4381799758388606683191116154, 0.66053553595983542267807889124, 2.38442427257391745338074240429, 3.17254494719466153093897900762, 3.77882129871347245680495548946, 4.629884908508955031249037531287, 5.28028142585524797152832977652, 6.28501873878640077802668055291, 6.795037528312349978330697873622, 7.60517047384070037569680537039, 8.30577584698516361657446205243, 8.725275883787480674590091673964, 9.809493891529843341990393791162, 10.87254606494202166974351557532, 11.41813731534558130964129014867, 12.143148868381191181155005488615, 12.94849164300056066084179679285, 13.4602167492081373096174775635, 14.35693241703948145630278868026, 15.09319719680174729692673223408, 15.35612677743702093775433137357, 16.149729185683847961584356047806, 16.89094722604630723607148443227, 17.404778678102897170880663081589, 18.31813787739627344377290508243, 18.92076336933543221308741175876

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