Properties

Label 2-3381-1.1-c1-0-100
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  + 0.443·2-s − 3-s − 1.80·4-s + 2.13·5-s − 0.443·6-s − 1.68·8-s + 9-s + 0.945·10-s − 4.24·11-s + 1.80·12-s − 2.11·13-s − 2.13·15-s + 2.85·16-s + 3.08·17-s + 0.443·18-s + 4.66·19-s − 3.84·20-s − 1.88·22-s + 23-s + 1.68·24-s − 0.458·25-s − 0.938·26-s − 27-s + 4.52·29-s − 0.945·30-s − 1.32·31-s + 4.64·32-s + ⋯
L(s)  = 1  + 0.313·2-s − 0.577·3-s − 0.901·4-s + 0.953·5-s − 0.181·6-s − 0.596·8-s + 0.333·9-s + 0.298·10-s − 1.28·11-s + 0.520·12-s − 0.586·13-s − 0.550·15-s + 0.714·16-s + 0.748·17-s + 0.104·18-s + 1.06·19-s − 0.859·20-s − 0.401·22-s + 0.208·23-s + 0.344·24-s − 0.0917·25-s − 0.184·26-s − 0.192·27-s + 0.839·29-s − 0.172·30-s − 0.238·31-s + 0.820·32-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 \( 1 - 0.443T + 2T^{2} \)
5 \( 1 - 2.13T + 5T^{2} \)
11 \( 1 + 4.24T + 11T^{2} \)
13 \( 1 + 2.11T + 13T^{2} \)
17 \( 1 - 3.08T + 17T^{2} \)
19 \( 1 - 4.66T + 19T^{2} \)
29 \( 1 - 4.52T + 29T^{2} \)
31 \( 1 + 1.32T + 31T^{2} \)
37 \( 1 - 0.106T + 37T^{2} \)
41 \( 1 + 2.90T + 41T^{2} \)
43 \( 1 + 6.11T + 43T^{2} \)
47 \( 1 + 9.37T + 47T^{2} \)
53 \( 1 - 7.00T + 53T^{2} \)
59 \( 1 + 10.7T + 59T^{2} \)
61 \( 1 + 7.02T + 61T^{2} \)
67 \( 1 + 14.9T + 67T^{2} \)
71 \( 1 - 4.38T + 71T^{2} \)
73 \( 1 - 14.9T + 73T^{2} \)
79 \( 1 - 13.0T + 79T^{2} \)
83 \( 1 + 5.97T + 83T^{2} \)
89 \( 1 + 0.438T + 89T^{2} \)
97 \( 1 + 3.31T + 97T^{2} \)
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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

−8.147785549819866536702010435393, −7.58603272628854128569453261787, −6.52334066939236275598058639502, −5.71203211858752640876701499984, −5.16825306238254259905780614099, −4.77263780449089704507687898525, −3.48725410505116672154394870218, −2.66561148352100372822327825150, −1.35059434317125230319504956231, 0, 1.35059434317125230319504956231, 2.66561148352100372822327825150, 3.48725410505116672154394870218, 4.77263780449089704507687898525, 5.16825306238254259905780614099, 5.71203211858752640876701499984, 6.52334066939236275598058639502, 7.58603272628854128569453261787, 8.147785549819866536702010435393

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