Properties

Label 2-3381-1.1-c1-0-46
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  − 0.593·2-s − 3-s − 1.64·4-s − 3.15·5-s + 0.593·6-s + 2.16·8-s + 9-s + 1.87·10-s − 6.54·11-s + 1.64·12-s − 0.692·13-s + 3.15·15-s + 2.01·16-s + 6.84·17-s − 0.593·18-s − 2.91·19-s + 5.19·20-s + 3.88·22-s − 23-s − 2.16·24-s + 4.93·25-s + 0.411·26-s − 27-s + 2.94·29-s − 1.87·30-s − 0.539·31-s − 5.52·32-s + ⋯
L(s)  = 1  − 0.419·2-s − 0.577·3-s − 0.823·4-s − 1.40·5-s + 0.242·6-s + 0.765·8-s + 0.333·9-s + 0.591·10-s − 1.97·11-s + 0.475·12-s − 0.192·13-s + 0.813·15-s + 0.502·16-s + 1.65·17-s − 0.139·18-s − 0.669·19-s + 1.16·20-s + 0.828·22-s − 0.208·23-s − 0.441·24-s + 0.986·25-s + 0.0806·26-s − 0.192·27-s + 0.547·29-s − 0.341·30-s − 0.0968·31-s − 0.976·32-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 \( 1 + 0.593T + 2T^{2} \)
5 \( 1 + 3.15T + 5T^{2} \)
11 \( 1 + 6.54T + 11T^{2} \)
13 \( 1 + 0.692T + 13T^{2} \)
17 \( 1 - 6.84T + 17T^{2} \)
19 \( 1 + 2.91T + 19T^{2} \)
29 \( 1 - 2.94T + 29T^{2} \)
31 \( 1 + 0.539T + 31T^{2} \)
37 \( 1 + 2.89T + 37T^{2} \)
41 \( 1 - 9.26T + 41T^{2} \)
43 \( 1 + 4.27T + 43T^{2} \)
47 \( 1 - 10.4T + 47T^{2} \)
53 \( 1 - 12.5T + 53T^{2} \)
59 \( 1 - 6.70T + 59T^{2} \)
61 \( 1 - 5.30T + 61T^{2} \)
67 \( 1 + 2.73T + 67T^{2} \)
71 \( 1 + 0.587T + 71T^{2} \)
73 \( 1 - 10.5T + 73T^{2} \)
79 \( 1 + 13.3T + 79T^{2} \)
83 \( 1 + 9.77T + 83T^{2} \)
89 \( 1 - 2.92T + 89T^{2} \)
97 \( 1 + 8.14T + 97T^{2} \)
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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

−8.143813841050541248232586718640, −7.65597370893142507880184389352, −7.14089597050558533898031020475, −5.67172465745383080962986156353, −5.24075294316133548430436228184, −4.36139186096881899251811901785, −3.70825862883724760151154719196, −2.60897179801659723337781183865, −0.881795175590712851918125535423, 0, 0.881795175590712851918125535423, 2.60897179801659723337781183865, 3.70825862883724760151154719196, 4.36139186096881899251811901785, 5.24075294316133548430436228184, 5.67172465745383080962986156353, 7.14089597050558533898031020475, 7.65597370893142507880184389352, 8.143813841050541248232586718640

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