Properties

Label 2-3381-1.1-c1-0-144
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  + 1.54·2-s + 3-s + 0.398·4-s + 1.02·5-s + 1.54·6-s − 2.48·8-s + 9-s + 1.58·10-s − 6.58·11-s + 0.398·12-s + 0.0967·13-s + 1.02·15-s − 4.63·16-s + 2.79·17-s + 1.54·18-s − 5.99·19-s + 0.408·20-s − 10.1·22-s + 23-s − 2.48·24-s − 3.94·25-s + 0.149·26-s + 27-s + 2.41·29-s + 1.58·30-s − 10.1·31-s − 2.22·32-s + ⋯
L(s)  = 1  + 1.09·2-s + 0.577·3-s + 0.199·4-s + 0.458·5-s + 0.632·6-s − 0.877·8-s + 0.333·9-s + 0.502·10-s − 1.98·11-s + 0.114·12-s + 0.0268·13-s + 0.264·15-s − 1.15·16-s + 0.677·17-s + 0.365·18-s − 1.37·19-s + 0.0913·20-s − 2.17·22-s + 0.208·23-s − 0.506·24-s − 0.789·25-s + 0.0293·26-s + 0.192·27-s + 0.447·29-s + 0.290·30-s − 1.81·31-s − 0.392·32-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 \( 1 - 1.54T + 2T^{2} \)
5 \( 1 - 1.02T + 5T^{2} \)
11 \( 1 + 6.58T + 11T^{2} \)
13 \( 1 - 0.0967T + 13T^{2} \)
17 \( 1 - 2.79T + 17T^{2} \)
19 \( 1 + 5.99T + 19T^{2} \)
29 \( 1 - 2.41T + 29T^{2} \)
31 \( 1 + 10.1T + 31T^{2} \)
37 \( 1 - 0.972T + 37T^{2} \)
41 \( 1 + 7.47T + 41T^{2} \)
43 \( 1 - 6.72T + 43T^{2} \)
47 \( 1 - 5.72T + 47T^{2} \)
53 \( 1 + 4.91T + 53T^{2} \)
59 \( 1 + 4.38T + 59T^{2} \)
61 \( 1 + 14.4T + 61T^{2} \)
67 \( 1 + 4.09T + 67T^{2} \)
71 \( 1 - 1.63T + 71T^{2} \)
73 \( 1 - 7.70T + 73T^{2} \)
79 \( 1 - 7.79T + 79T^{2} \)
83 \( 1 - 12.4T + 83T^{2} \)
89 \( 1 - 2.59T + 89T^{2} \)
97 \( 1 - 0.921T + 97T^{2} \)
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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

−8.112250219250831334825761216459, −7.59971131385375648967264640475, −6.52456757089302438950874527327, −5.70808007402360538255714998335, −5.20055495861619986182085116480, −4.39281101903202512743102028716, −3.50433591118086768680096856838, −2.71872806474941342819360105742, −1.99026214723177515932425598026, 0, 1.99026214723177515932425598026, 2.71872806474941342819360105742, 3.50433591118086768680096856838, 4.39281101903202512743102028716, 5.20055495861619986182085116480, 5.70808007402360538255714998335, 6.52456757089302438950874527327, 7.59971131385375648967264640475, 8.112250219250831334825761216459

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