Properties

Label 2-3381-1.1-c1-0-77
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  − 1.83·2-s − 3-s + 1.35·4-s − 0.943·5-s + 1.83·6-s + 1.17·8-s + 9-s + 1.72·10-s + 2.08·11-s − 1.35·12-s + 3.64·13-s + 0.943·15-s − 4.87·16-s + 0.299·17-s − 1.83·18-s − 0.579·19-s − 1.28·20-s − 3.82·22-s − 23-s − 1.17·24-s − 4.10·25-s − 6.67·26-s − 27-s − 5.89·29-s − 1.72·30-s − 6.02·31-s + 6.57·32-s + ⋯
L(s)  = 1  − 1.29·2-s − 0.577·3-s + 0.679·4-s − 0.421·5-s + 0.748·6-s + 0.415·8-s + 0.333·9-s + 0.546·10-s + 0.629·11-s − 0.392·12-s + 1.01·13-s + 0.243·15-s − 1.21·16-s + 0.0726·17-s − 0.431·18-s − 0.132·19-s − 0.286·20-s − 0.815·22-s − 0.208·23-s − 0.239·24-s − 0.821·25-s − 1.30·26-s − 0.192·27-s − 1.09·29-s − 0.315·30-s − 1.08·31-s + 1.16·32-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 \( 1 + 1.83T + 2T^{2} \)
5 \( 1 + 0.943T + 5T^{2} \)
11 \( 1 - 2.08T + 11T^{2} \)
13 \( 1 - 3.64T + 13T^{2} \)
17 \( 1 - 0.299T + 17T^{2} \)
19 \( 1 + 0.579T + 19T^{2} \)
29 \( 1 + 5.89T + 29T^{2} \)
31 \( 1 + 6.02T + 31T^{2} \)
37 \( 1 + 1.23T + 37T^{2} \)
41 \( 1 - 8.42T + 41T^{2} \)
43 \( 1 + 9.08T + 43T^{2} \)
47 \( 1 - 5.06T + 47T^{2} \)
53 \( 1 - 2.32T + 53T^{2} \)
59 \( 1 - 2.08T + 59T^{2} \)
61 \( 1 - 12.4T + 61T^{2} \)
67 \( 1 + 4.13T + 67T^{2} \)
71 \( 1 + 11.6T + 71T^{2} \)
73 \( 1 + 11.5T + 73T^{2} \)
79 \( 1 - 17.1T + 79T^{2} \)
83 \( 1 - 10.9T + 83T^{2} \)
89 \( 1 - 2.67T + 89T^{2} \)
97 \( 1 - 0.589T + 97T^{2} \)
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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

−8.298289550237837879499007577732, −7.62071676396072412836779425052, −6.98203143622846150692232954392, −6.15671674997109005187509826057, −5.36485949801935358931643735180, −4.20362203903959719944076108976, −3.66200637203610797145064834959, −2.04086540020696863110292642239, −1.14868214418247025152375760072, 0, 1.14868214418247025152375760072, 2.04086540020696863110292642239, 3.66200637203610797145064834959, 4.20362203903959719944076108976, 5.36485949801935358931643735180, 6.15671674997109005187509826057, 6.98203143622846150692232954392, 7.62071676396072412836779425052, 8.298289550237837879499007577732

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