Properties

Label 2-3420-1.1-c1-0-4
Degree $2$
Conductor $3420$
Sign $1$
Analytic cond. $27.3088$
Root an. cond. $5.22578$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 2·7-s − 3.46·11-s + 0.732·13-s − 3.46·17-s + 19-s − 3.46·23-s + 25-s + 3.46·29-s + 5.46·31-s − 2·35-s + 3.26·37-s + 6·41-s + 8.92·43-s + 0.928·47-s − 3·49-s + 7.26·53-s + 3.46·55-s + 6.92·59-s − 8.39·61-s − 0.732·65-s + 3.26·67-s + 9.46·71-s − 7.46·73-s − 6.92·77-s − 10.9·79-s + 3.46·83-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.755·7-s − 1.04·11-s + 0.203·13-s − 0.840·17-s + 0.229·19-s − 0.722·23-s + 0.200·25-s + 0.643·29-s + 0.981·31-s − 0.338·35-s + 0.537·37-s + 0.937·41-s + 1.36·43-s + 0.135·47-s − 0.428·49-s + 0.998·53-s + 0.467·55-s + 0.901·59-s − 1.07·61-s − 0.0907·65-s + 0.399·67-s + 1.12·71-s − 0.873·73-s − 0.789·77-s − 1.22·79-s + 0.380·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3420\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $1$
Analytic conductor: \(27.3088\)
Root analytic conductor: \(5.22578\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3420,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.627094175\)
\(L(\frac12)\) \(\approx\) \(1.627094175\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + T \)
19 \( 1 - T \)
good7 \( 1 - 2T + 7T^{2} \)
11 \( 1 + 3.46T + 11T^{2} \)
13 \( 1 - 0.732T + 13T^{2} \)
17 \( 1 + 3.46T + 17T^{2} \)
23 \( 1 + 3.46T + 23T^{2} \)
29 \( 1 - 3.46T + 29T^{2} \)
31 \( 1 - 5.46T + 31T^{2} \)
37 \( 1 - 3.26T + 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 8.92T + 43T^{2} \)
47 \( 1 - 0.928T + 47T^{2} \)
53 \( 1 - 7.26T + 53T^{2} \)
59 \( 1 - 6.92T + 59T^{2} \)
61 \( 1 + 8.39T + 61T^{2} \)
67 \( 1 - 3.26T + 67T^{2} \)
71 \( 1 - 9.46T + 71T^{2} \)
73 \( 1 + 7.46T + 73T^{2} \)
79 \( 1 + 10.9T + 79T^{2} \)
83 \( 1 - 3.46T + 83T^{2} \)
89 \( 1 - 8.53T + 89T^{2} \)
97 \( 1 - 14.5T + 97T^{2} \)
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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

−8.419560780479533498674393881632, −7.927790483488419868314294261614, −7.29615915720755471268165866780, −6.33362801843414492216741919866, −5.54370577975522310387811567902, −4.66121459892390849198714665659, −4.13223568276784652380286276395, −2.91433520063197880129429776464, −2.12514771523704811962369045359, −0.75052375508716893642255794850, 0.75052375508716893642255794850, 2.12514771523704811962369045359, 2.91433520063197880129429776464, 4.13223568276784652380286276395, 4.66121459892390849198714665659, 5.54370577975522310387811567902, 6.33362801843414492216741919866, 7.29615915720755471268165866780, 7.927790483488419868314294261614, 8.419560780479533498674393881632

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