Properties

Label 2-3381-1.1-c1-0-103
Degree $2$
Conductor $3381$
Sign $-1$
Analytic cond. $26.9974$
Root an. cond. $5.19590$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.289·2-s − 3-s − 1.91·4-s + 2.32·5-s − 0.289·6-s − 1.13·8-s + 9-s + 0.674·10-s + 0.730·11-s + 1.91·12-s − 0.115·13-s − 2.32·15-s + 3.50·16-s + 0.789·17-s + 0.289·18-s − 6.92·19-s − 4.45·20-s + 0.211·22-s − 23-s + 1.13·24-s + 0.411·25-s − 0.0334·26-s − 27-s − 6.26·29-s − 0.674·30-s + 1.49·31-s + 3.28·32-s + ⋯
L(s)  = 1  + 0.205·2-s − 0.577·3-s − 0.957·4-s + 1.04·5-s − 0.118·6-s − 0.401·8-s + 0.333·9-s + 0.213·10-s + 0.220·11-s + 0.553·12-s − 0.0319·13-s − 0.600·15-s + 0.875·16-s + 0.191·17-s + 0.0683·18-s − 1.58·19-s − 0.996·20-s + 0.0451·22-s − 0.208·23-s + 0.231·24-s + 0.0822·25-s − 0.00655·26-s − 0.192·27-s − 1.16·29-s − 0.123·30-s + 0.268·31-s + 0.580·32-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $-1$
Analytic conductor: \(26.9974\)
Root analytic conductor: \(5.19590\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3381,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
7 \( 1 \)
23 \( 1 + T \)
good2 \( 1 - 0.289T + 2T^{2} \)
5 \( 1 - 2.32T + 5T^{2} \)
11 \( 1 - 0.730T + 11T^{2} \)
13 \( 1 + 0.115T + 13T^{2} \)
17 \( 1 - 0.789T + 17T^{2} \)
19 \( 1 + 6.92T + 19T^{2} \)
29 \( 1 + 6.26T + 29T^{2} \)
31 \( 1 - 1.49T + 31T^{2} \)
37 \( 1 - 4.14T + 37T^{2} \)
41 \( 1 - 8.20T + 41T^{2} \)
43 \( 1 - 1.50T + 43T^{2} \)
47 \( 1 + 3.78T + 47T^{2} \)
53 \( 1 + 6.81T + 53T^{2} \)
59 \( 1 + 6.18T + 59T^{2} \)
61 \( 1 + 2.39T + 61T^{2} \)
67 \( 1 + 8.37T + 67T^{2} \)
71 \( 1 + 1.53T + 71T^{2} \)
73 \( 1 - 9.90T + 73T^{2} \)
79 \( 1 - 0.483T + 79T^{2} \)
83 \( 1 + 1.84T + 83T^{2} \)
89 \( 1 - 3.06T + 89T^{2} \)
97 \( 1 - 14.8T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.297206605499430325939870204626, −7.53934038596664096513271068472, −6.27579435644903942979943908471, −6.08002592368892117591924807038, −5.21087731229118115730003317761, −4.46877088050710787955109288420, −3.74804253119229425166022467574, −2.46037526772057143821177388677, −1.39980444621589590491967583280, 0, 1.39980444621589590491967583280, 2.46037526772057143821177388677, 3.74804253119229425166022467574, 4.46877088050710787955109288420, 5.21087731229118115730003317761, 6.08002592368892117591924807038, 6.27579435644903942979943908471, 7.53934038596664096513271068472, 8.297206605499430325939870204626

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