Properties

Label 2-3380-1.1-c1-0-36
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.60·3-s + 5-s + 2.76·7-s + 3.76·9-s + 2.16·11-s + 2.60·15-s + 5.03·19-s + 7.20·21-s + 4.93·23-s + 25-s + 2.00·27-s − 4.43·29-s − 3.37·31-s + 5.63·33-s + 2.76·35-s − 3.97·37-s − 1.66·41-s − 9.80·43-s + 3.76·45-s + 11.6·47-s + 0.665·49-s − 12.7·53-s + 2.16·55-s + 13.1·57-s − 8.16·59-s − 10.3·61-s + 10.4·63-s + ⋯
L(s)  = 1  + 1.50·3-s + 0.447·5-s + 1.04·7-s + 1.25·9-s + 0.653·11-s + 0.671·15-s + 1.15·19-s + 1.57·21-s + 1.02·23-s + 0.200·25-s + 0.384·27-s − 0.823·29-s − 0.605·31-s + 0.981·33-s + 0.468·35-s − 0.653·37-s − 0.260·41-s − 1.49·43-s + 0.561·45-s + 1.69·47-s + 0.0951·49-s − 1.75·53-s + 0.292·55-s + 1.73·57-s − 1.06·59-s − 1.31·61-s + 1.31·63-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: \(0\)
Selberg data: \((2,\ 3380,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.333875163\)
\(L(\frac12)\) \(\approx\) \(4.333875163\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.60T + 3T^{2} \)
7 \( 1 - 2.76T + 7T^{2} \)
11 \( 1 - 2.16T + 11T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 5.03T + 19T^{2} \)
23 \( 1 - 4.93T + 23T^{2} \)
29 \( 1 + 4.43T + 29T^{2} \)
31 \( 1 + 3.37T + 31T^{2} \)
37 \( 1 + 3.97T + 37T^{2} \)
41 \( 1 + 1.66T + 41T^{2} \)
43 \( 1 + 9.80T + 43T^{2} \)
47 \( 1 - 11.6T + 47T^{2} \)
53 \( 1 + 12.7T + 53T^{2} \)
59 \( 1 + 8.16T + 59T^{2} \)
61 \( 1 + 10.3T + 61T^{2} \)
67 \( 1 + 7.10T + 67T^{2} \)
71 \( 1 - 16.1T + 71T^{2} \)
73 \( 1 - 15.9T + 73T^{2} \)
79 \( 1 - 11.9T + 79T^{2} \)
83 \( 1 + 3.97T + 83T^{2} \)
89 \( 1 + 7.94T + 89T^{2} \)
97 \( 1 - 0.462T + 97T^{2} \)
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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

−8.673952038725713576442875211455, −7.901225619490832545047571138817, −7.41391524767570049627124251575, −6.56095157822754553106056105817, −5.40823499143385000683210246605, −4.75297502228793133709639180381, −3.68381993221106659668510947671, −3.07341322162719367414542733606, −1.98809386082915146684609802538, −1.36103245414624718849959780079, 1.36103245414624718849959780079, 1.98809386082915146684609802538, 3.07341322162719367414542733606, 3.68381993221106659668510947671, 4.75297502228793133709639180381, 5.40823499143385000683210246605, 6.56095157822754553106056105817, 7.41391524767570049627124251575, 7.901225619490832545047571138817, 8.673952038725713576442875211455

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