Properties

Label 2-3380-1.1-c1-0-41
Degree $2$
Conductor $3380$
Sign $-1$
Analytic cond. $26.9894$
Root an. cond. $5.19513$
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  − 0.339·3-s − 5-s + 3.88·7-s − 2.88·9-s + 1.54·11-s + 0.339·15-s − 2.86·19-s − 1.32·21-s − 5.42·23-s + 25-s + 2·27-s − 5.20·29-s − 6.22·31-s − 0.524·33-s − 3.88·35-s − 8.56·37-s + 9.08·41-s − 0.980·43-s + 2.88·45-s − 6.52·47-s + 8.08·49-s + 6.44·53-s − 1.54·55-s + 0.973·57-s + 4.45·59-s + 9.65·61-s − 11.2·63-s + ⋯
L(s)  = 1  − 0.196·3-s − 0.447·5-s + 1.46·7-s − 0.961·9-s + 0.465·11-s + 0.0877·15-s − 0.657·19-s − 0.288·21-s − 1.13·23-s + 0.200·25-s + 0.384·27-s − 0.966·29-s − 1.11·31-s − 0.0913·33-s − 0.656·35-s − 1.40·37-s + 1.41·41-s − 0.149·43-s + 0.429·45-s − 0.951·47-s + 1.15·49-s + 0.885·53-s − 0.208·55-s + 0.128·57-s + 0.580·59-s + 1.23·61-s − 1.41·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3380\)    =    \(2^{2} \cdot 5 \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(26.9894\)
Root analytic conductor: \(5.19513\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3380} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3380,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 \)
5 \( 1 + T \)
13 \( 1 \)
good3 \( 1 + 0.339T + 3T^{2} \)
7 \( 1 - 3.88T + 7T^{2} \)
11 \( 1 - 1.54T + 11T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 2.86T + 19T^{2} \)
23 \( 1 + 5.42T + 23T^{2} \)
29 \( 1 + 5.20T + 29T^{2} \)
31 \( 1 + 6.22T + 31T^{2} \)
37 \( 1 + 8.56T + 37T^{2} \)
41 \( 1 - 9.08T + 41T^{2} \)
43 \( 1 + 0.980T + 43T^{2} \)
47 \( 1 + 6.52T + 47T^{2} \)
53 \( 1 - 6.44T + 53T^{2} \)
59 \( 1 - 4.45T + 59T^{2} \)
61 \( 1 - 9.65T + 61T^{2} \)
67 \( 1 + 6.97T + 67T^{2} \)
71 \( 1 - 12.6T + 71T^{2} \)
73 \( 1 + 3.43T + 73T^{2} \)
79 \( 1 + 13.1T + 79T^{2} \)
83 \( 1 + 8.56T + 83T^{2} \)
89 \( 1 + 17.1T + 89T^{2} \)
97 \( 1 + 13.7T + 97T^{2} \)
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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

−8.377152651017062553437221900727, −7.58964389050066241456294460628, −6.84145696827528899800832713178, −5.76907972313469933364996967159, −5.30773643841967203561073081225, −4.31255494357185134429211568235, −3.69538610509362182223665025484, −2.39953244332321999250912184705, −1.51780181829343858783964529183, 0, 1.51780181829343858783964529183, 2.39953244332321999250912184705, 3.69538610509362182223665025484, 4.31255494357185134429211568235, 5.30773643841967203561073081225, 5.76907972313469933364996967159, 6.84145696827528899800832713178, 7.58964389050066241456294460628, 8.377152651017062553437221900727

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