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Properties

Label 1-3381-3381.1019-r0-0-0
Degree $1$
Conductor $3381$
Sign $0.291 - 0.956i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.291 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.291 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.291 - 0.956i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (1019, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ 0.291 - 0.956i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.017457556 - 0.7537731032i\)
\(L(\frac12)\) \(\approx\) \(1.017457556 - 0.7537731032i\)
\(L(1)\) \(\approx\) \(0.9486783133 + 0.03264639398i\)
\(L(1)\) \(\approx\) \(0.9486783133 + 0.03264639398i\)

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

−19.09702557170877530343404999373, −18.22895688205374334894163230100, −17.83519511228224489673042123156, −17.20187517811366428322718662499, −16.36260122525705524843942625136, −15.31909179911380516738545780116, −14.61589065580829913181246715565, −14.06273763158844820820022400648, −13.18543813220337071810403042125, −12.79978657371084185052876659249, −11.8585380217290772853972390357, −11.10189883847134946939099481956, −10.61588398949073543832318754298, −9.942449137569204353441741082674, −9.26506629572579050593992676740, −8.585319085495531503863831325979, −7.57877518060942877781535365270, −6.90726089017518090973847577413, −5.99278733447679326706081583957, −5.1056808304557953869520661570, −4.270541338922280461534776586099, −3.53851379053671574055949210766, −2.56714083509642649801989995455, −2.08655189616768023764778393137, −1.213615045204170276670985811180, 0.45077024432983540092797120212, 1.14624605817097271885474976908, 2.42606145001812730446398788014, 3.48208805136938347730586408388, 4.42414127736229707182950715254, 5.15266207755715608021600977241, 5.9632979948789136386742767218, 6.08964767123792460965294871747, 7.52832371876123894097210472531, 7.98738833400676714253182212052, 8.58884053000953423874996847124, 9.466985265917943018307807056710, 9.93453391324811447645927545309, 10.78164691532242515894717125149, 11.77333883457071958201181453496, 12.78605423535230266738366481997, 13.24871538721123069862534512582, 13.77258167867999809565805528485, 14.593690257051711613088038925289, 15.336864439624872770585392175531, 16.14818164914955624060106326901, 16.48372367459638641663143078923, 17.14849710008844768402086951073, 18.00116605449117240453852990868, 18.374199950410164196804045887046

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