Properties

Label 2-3381-1.1-c1-0-105
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  − 2.27·2-s − 3-s + 3.16·4-s + 4.15·5-s + 2.27·6-s − 2.64·8-s + 9-s − 9.43·10-s − 3.18·11-s − 3.16·12-s + 0.175·13-s − 4.15·15-s − 0.312·16-s − 1.32·17-s − 2.27·18-s + 0.0504·19-s + 13.1·20-s + 7.24·22-s − 23-s + 2.64·24-s + 12.2·25-s − 0.399·26-s − 27-s + 1.10·29-s + 9.43·30-s − 8.71·31-s + 6.00·32-s + ⋯
L(s)  = 1  − 1.60·2-s − 0.577·3-s + 1.58·4-s + 1.85·5-s + 0.927·6-s − 0.936·8-s + 0.333·9-s − 2.98·10-s − 0.960·11-s − 0.913·12-s + 0.0487·13-s − 1.07·15-s − 0.0781·16-s − 0.321·17-s − 0.535·18-s + 0.0115·19-s + 2.93·20-s + 1.54·22-s − 0.208·23-s + 0.540·24-s + 2.44·25-s − 0.0782·26-s − 0.192·27-s + 0.204·29-s + 1.72·30-s − 1.56·31-s + 1.06·32-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: $\chi_{3381} (1, \cdot )$
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 \( 1 + 2.27T + 2T^{2} \)
5 \( 1 - 4.15T + 5T^{2} \)
11 \( 1 + 3.18T + 11T^{2} \)
13 \( 1 - 0.175T + 13T^{2} \)
17 \( 1 + 1.32T + 17T^{2} \)
19 \( 1 - 0.0504T + 19T^{2} \)
29 \( 1 - 1.10T + 29T^{2} \)
31 \( 1 + 8.71T + 31T^{2} \)
37 \( 1 - 3.76T + 37T^{2} \)
41 \( 1 - 4.84T + 41T^{2} \)
43 \( 1 + 10.5T + 43T^{2} \)
47 \( 1 + 8.95T + 47T^{2} \)
53 \( 1 + 9.97T + 53T^{2} \)
59 \( 1 + 6.38T + 59T^{2} \)
61 \( 1 + 7.23T + 61T^{2} \)
67 \( 1 - 6.02T + 67T^{2} \)
71 \( 1 - 6.71T + 71T^{2} \)
73 \( 1 + 4.61T + 73T^{2} \)
79 \( 1 + 15.4T + 79T^{2} \)
83 \( 1 - 7.21T + 83T^{2} \)
89 \( 1 + 1.94T + 89T^{2} \)
97 \( 1 + 12.7T + 97T^{2} \)
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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

−8.399374464540835844339079505473, −7.61327429962012431652466248934, −6.76851029260870129518543365656, −6.19840497704583980206790323790, −5.45084775834991170161490397124, −4.70936889141501132749456366423, −2.95663934279639995700306735885, −2.02773190174070506546621259357, −1.41161963702415896113339356655, 0, 1.41161963702415896113339356655, 2.02773190174070506546621259357, 2.95663934279639995700306735885, 4.70936889141501132749456366423, 5.45084775834991170161490397124, 6.19840497704583980206790323790, 6.76851029260870129518543365656, 7.61327429962012431652466248934, 8.399374464540835844339079505473

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