Properties

Label 2-3420-1.1-c1-0-8
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 − 2.60·7-s + 4.60·11-s + 4.60·13-s − 2·17-s − 19-s + 2·23-s + 25-s − 2.60·29-s + 4·31-s − 2.60·35-s + 3.39·37-s − 6.60·41-s + 10.6·43-s − 6·47-s − 0.211·49-s + 4.60·55-s − 5.21·59-s − 7.21·61-s + 4.60·65-s + 4·67-s + 9.21·71-s + 6·73-s − 12·77-s + 8·79-s + 11.2·83-s − 2·85-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.984·7-s + 1.38·11-s + 1.27·13-s − 0.485·17-s − 0.229·19-s + 0.417·23-s + 0.200·25-s − 0.483·29-s + 0.718·31-s − 0.440·35-s + 0.558·37-s − 1.03·41-s + 1.61·43-s − 0.875·47-s − 0.0301·49-s + 0.621·55-s − 0.678·59-s − 0.923·61-s + 0.571·65-s + 0.488·67-s + 1.09·71-s + 0.702·73-s − 1.36·77-s + 0.900·79-s + 1.23·83-s − 0.216·85-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: $\chi_{3420} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3420,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.080510788\)
\(L(\frac12)\) \(\approx\) \(2.080510788\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.60T + 7T^{2} \)
11 \( 1 - 4.60T + 11T^{2} \)
13 \( 1 - 4.60T + 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
23 \( 1 - 2T + 23T^{2} \)
29 \( 1 + 2.60T + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 - 3.39T + 37T^{2} \)
41 \( 1 + 6.60T + 41T^{2} \)
43 \( 1 - 10.6T + 43T^{2} \)
47 \( 1 + 6T + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 5.21T + 59T^{2} \)
61 \( 1 + 7.21T + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 - 9.21T + 71T^{2} \)
73 \( 1 - 6T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 11.2T + 83T^{2} \)
89 \( 1 + 6.60T + 89T^{2} \)
97 \( 1 - 16.6T + 97T^{2} \)
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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

−8.822585385534432734032987068424, −7.928708076252259009844733481865, −6.78433564038838064896394215827, −6.41654403290332542738170326728, −5.86640660606753206783373254575, −4.69340181874240815734866225166, −3.80744439581216848878985276671, −3.17811019529864423021984349203, −1.95759953865122517374385118595, −0.887692011863933755740100291764, 0.887692011863933755740100291764, 1.95759953865122517374385118595, 3.17811019529864423021984349203, 3.80744439581216848878985276671, 4.69340181874240815734866225166, 5.86640660606753206783373254575, 6.41654403290332542738170326728, 6.78433564038838064896394215827, 7.928708076252259009844733481865, 8.822585385534432734032987068424

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