Properties

Label 2-3381-1.1-c1-0-70
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  − 1.61·2-s − 3-s + 0.618·4-s − 1.38·5-s + 1.61·6-s + 2.23·8-s + 9-s + 2.23·10-s + 11-s − 0.618·12-s + 1.61·13-s + 1.38·15-s − 4.85·16-s − 3.47·17-s − 1.61·18-s + 0.236·19-s − 0.854·20-s − 1.61·22-s + 23-s − 2.23·24-s − 3.09·25-s − 2.61·26-s − 27-s + 6.23·29-s − 2.23·30-s − 4.70·31-s + 3.38·32-s + ⋯
L(s)  = 1  − 1.14·2-s − 0.577·3-s + 0.309·4-s − 0.618·5-s + 0.660·6-s + 0.790·8-s + 0.333·9-s + 0.707·10-s + 0.301·11-s − 0.178·12-s + 0.448·13-s + 0.356·15-s − 1.21·16-s − 0.842·17-s − 0.381·18-s + 0.0541·19-s − 0.190·20-s − 0.344·22-s + 0.208·23-s − 0.456·24-s − 0.618·25-s − 0.513·26-s − 0.192·27-s + 1.15·29-s − 0.408·30-s − 0.845·31-s + 0.597·32-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 \( 1 + 1.61T + 2T^{2} \)
5 \( 1 + 1.38T + 5T^{2} \)
11 \( 1 - T + 11T^{2} \)
13 \( 1 - 1.61T + 13T^{2} \)
17 \( 1 + 3.47T + 17T^{2} \)
19 \( 1 - 0.236T + 19T^{2} \)
29 \( 1 - 6.23T + 29T^{2} \)
31 \( 1 + 4.70T + 31T^{2} \)
37 \( 1 + 4.23T + 37T^{2} \)
41 \( 1 + 5.47T + 41T^{2} \)
43 \( 1 - 2.85T + 43T^{2} \)
47 \( 1 - 1.70T + 47T^{2} \)
53 \( 1 - 11.0T + 53T^{2} \)
59 \( 1 - 1.38T + 59T^{2} \)
61 \( 1 - 1.14T + 61T^{2} \)
67 \( 1 - 3.09T + 67T^{2} \)
71 \( 1 - 5.32T + 71T^{2} \)
73 \( 1 + 6.23T + 73T^{2} \)
79 \( 1 + 9.76T + 79T^{2} \)
83 \( 1 - 0.0557T + 83T^{2} \)
89 \( 1 - 6.56T + 89T^{2} \)
97 \( 1 - 6.70T + 97T^{2} \)
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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

−8.501954898033375761711308173917, −7.51543241308337602333079311559, −7.00939397559259908016658271317, −6.20104465273456098967994770912, −5.16756878803702128324797287319, −4.35600052022758166203156529886, −3.62944497746666368708655234986, −2.15906198549891827401582384598, −1.06787939464176201037385868809, 0, 1.06787939464176201037385868809, 2.15906198549891827401582384598, 3.62944497746666368708655234986, 4.35600052022758166203156529886, 5.16756878803702128324797287319, 6.20104465273456098967994770912, 7.00939397559259908016658271317, 7.51543241308337602333079311559, 8.501954898033375761711308173917

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