Properties

Label 2-3381-1.1-c1-0-139
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  + 2.48·2-s − 3-s + 4.18·4-s − 2.95·5-s − 2.48·6-s + 5.44·8-s + 9-s − 7.34·10-s − 0.300·11-s − 4.18·12-s − 3.25·13-s + 2.95·15-s + 5.15·16-s + 2.43·17-s + 2.48·18-s + 0.970·19-s − 12.3·20-s − 0.747·22-s + 23-s − 5.44·24-s + 3.71·25-s − 8.09·26-s − 27-s − 6.80·29-s + 7.34·30-s − 2.23·31-s + 1.94·32-s + ⋯
L(s)  = 1  + 1.75·2-s − 0.577·3-s + 2.09·4-s − 1.32·5-s − 1.01·6-s + 1.92·8-s + 0.333·9-s − 2.32·10-s − 0.0905·11-s − 1.20·12-s − 0.902·13-s + 0.762·15-s + 1.28·16-s + 0.590·17-s + 0.586·18-s + 0.222·19-s − 2.76·20-s − 0.159·22-s + 0.208·23-s − 1.11·24-s + 0.743·25-s − 1.58·26-s − 0.192·27-s − 1.26·29-s + 1.34·30-s − 0.401·31-s + 0.344·32-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: Trivial
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 \( 1 - 2.48T + 2T^{2} \)
5 \( 1 + 2.95T + 5T^{2} \)
11 \( 1 + 0.300T + 11T^{2} \)
13 \( 1 + 3.25T + 13T^{2} \)
17 \( 1 - 2.43T + 17T^{2} \)
19 \( 1 - 0.970T + 19T^{2} \)
29 \( 1 + 6.80T + 29T^{2} \)
31 \( 1 + 2.23T + 31T^{2} \)
37 \( 1 + 11.4T + 37T^{2} \)
41 \( 1 + 0.769T + 41T^{2} \)
43 \( 1 + 7.25T + 43T^{2} \)
47 \( 1 - 4.88T + 47T^{2} \)
53 \( 1 + 3.72T + 53T^{2} \)
59 \( 1 + 8.22T + 59T^{2} \)
61 \( 1 - 7.13T + 61T^{2} \)
67 \( 1 + 1.47T + 67T^{2} \)
71 \( 1 + 1.85T + 71T^{2} \)
73 \( 1 + 13.9T + 73T^{2} \)
79 \( 1 - 13.1T + 79T^{2} \)
83 \( 1 + 9.41T + 83T^{2} \)
89 \( 1 + 7.19T + 89T^{2} \)
97 \( 1 + 1.43T + 97T^{2} \)
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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

−7.72879885669060147781257494042, −7.30412295599284880828537382502, −6.66824416332416588297364816012, −5.62510779053437322607366459755, −5.14097316226701265655770015584, −4.40020310846032748303835121093, −3.67117293685936556768412915123, −3.06197372247463101654167557982, −1.78958832044571937198890335764, 0, 1.78958832044571937198890335764, 3.06197372247463101654167557982, 3.67117293685936556768412915123, 4.40020310846032748303835121093, 5.14097316226701265655770015584, 5.62510779053437322607366459755, 6.66824416332416588297364816012, 7.30412295599284880828537382502, 7.72879885669060147781257494042

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