Properties

Label 2-3380-1.1-c1-0-13
Degree $2$
Conductor $3380$
Sign $1$
Analytic cond. $26.9894$
Root an. cond. $5.19513$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s + 7-s − 2·9-s − 3·11-s − 15-s − 3·17-s + 7·19-s + 21-s − 3·23-s + 25-s − 5·27-s + 3·29-s + 4·31-s − 3·33-s − 35-s + 7·37-s + 9·41-s + 11·43-s + 2·45-s − 6·49-s − 3·51-s − 6·53-s + 3·55-s + 7·57-s + 3·59-s + 11·61-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 0.377·7-s − 2/3·9-s − 0.904·11-s − 0.258·15-s − 0.727·17-s + 1.60·19-s + 0.218·21-s − 0.625·23-s + 1/5·25-s − 0.962·27-s + 0.557·29-s + 0.718·31-s − 0.522·33-s − 0.169·35-s + 1.15·37-s + 1.40·41-s + 1.67·43-s + 0.298·45-s − 6/7·49-s − 0.420·51-s − 0.824·53-s + 0.404·55-s + 0.927·57-s + 0.390·59-s + 1.40·61-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: yes
Arithmetic: yes
Character: $\chi_{3380} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3380,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.858053926\)
\(L(\frac12)\) \(\approx\) \(1.858053926\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 - 11 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.437466067394014803273268723976, −7.900762301908805831018863224726, −7.46472538303385328714100229337, −6.34385109093521830852023641926, −5.53495183322265117333042594954, −4.76049910203042724723382303652, −3.87506980248141919150627784445, −2.89641602178000814777213771695, −2.31992354194607079312415716352, −0.77395588808192248890668110143, 0.77395588808192248890668110143, 2.31992354194607079312415716352, 2.89641602178000814777213771695, 3.87506980248141919150627784445, 4.76049910203042724723382303652, 5.53495183322265117333042594954, 6.34385109093521830852023641926, 7.46472538303385328714100229337, 7.900762301908805831018863224726, 8.437466067394014803273268723976

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