Properties

Label 2-3381-1.1-c1-0-140
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  + 2.04·2-s − 3-s + 2.17·4-s + 1.94·5-s − 2.04·6-s + 0.358·8-s + 9-s + 3.97·10-s − 5.66·11-s − 2.17·12-s − 6.17·13-s − 1.94·15-s − 3.61·16-s + 7.23·17-s + 2.04·18-s + 0.790·19-s + 4.22·20-s − 11.5·22-s − 23-s − 0.358·24-s − 1.22·25-s − 12.6·26-s − 27-s − 5.34·29-s − 3.97·30-s + 0.911·31-s − 8.11·32-s + ⋯
L(s)  = 1  + 1.44·2-s − 0.577·3-s + 1.08·4-s + 0.869·5-s − 0.834·6-s + 0.126·8-s + 0.333·9-s + 1.25·10-s − 1.70·11-s − 0.628·12-s − 1.71·13-s − 0.501·15-s − 0.904·16-s + 1.75·17-s + 0.481·18-s + 0.181·19-s + 0.945·20-s − 2.46·22-s − 0.208·23-s − 0.0731·24-s − 0.244·25-s − 2.47·26-s − 0.192·27-s − 0.991·29-s − 0.725·30-s + 0.163·31-s − 1.43·32-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 \( 1 - 2.04T + 2T^{2} \)
5 \( 1 - 1.94T + 5T^{2} \)
11 \( 1 + 5.66T + 11T^{2} \)
13 \( 1 + 6.17T + 13T^{2} \)
17 \( 1 - 7.23T + 17T^{2} \)
19 \( 1 - 0.790T + 19T^{2} \)
29 \( 1 + 5.34T + 29T^{2} \)
31 \( 1 - 0.911T + 31T^{2} \)
37 \( 1 + 0.973T + 37T^{2} \)
41 \( 1 + 1.51T + 41T^{2} \)
43 \( 1 + 2.70T + 43T^{2} \)
47 \( 1 + 1.52T + 47T^{2} \)
53 \( 1 + 8.95T + 53T^{2} \)
59 \( 1 + 10.8T + 59T^{2} \)
61 \( 1 + 6.31T + 61T^{2} \)
67 \( 1 - 7.31T + 67T^{2} \)
71 \( 1 + 8.75T + 71T^{2} \)
73 \( 1 + 0.875T + 73T^{2} \)
79 \( 1 - 5.96T + 79T^{2} \)
83 \( 1 + 0.273T + 83T^{2} \)
89 \( 1 - 17.6T + 89T^{2} \)
97 \( 1 - 3.09T + 97T^{2} \)
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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

−7.70363736735894027321237187827, −7.52545010086572394530187033578, −6.29610044989852776907719575087, −5.74048925018908034862626084717, −5.03280564333685543035445025294, −4.89617029917841762770179652193, −3.50073678803794701649485819939, −2.73354199150103204961218534721, −1.90329395284133856898625048937, 0, 1.90329395284133856898625048937, 2.73354199150103204961218534721, 3.50073678803794701649485819939, 4.89617029917841762770179652193, 5.03280564333685543035445025294, 5.74048925018908034862626084717, 6.29610044989852776907719575087, 7.52545010086572394530187033578, 7.70363736735894027321237187827

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