Properties

Label 2-3920-1.1-c1-0-34
Degree $2$
Conductor $3920$
Sign $-1$
Analytic cond. $31.3013$
Root an. cond. $5.59476$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2.56·3-s − 5-s + 3.56·9-s − 2.56·11-s − 4.56·13-s + 2.56·15-s + 4.56·17-s + 1.12·19-s + 5.12·23-s + 25-s − 1.43·27-s − 5.68·29-s + 6.56·33-s + 6·37-s + 11.6·39-s + 3.12·41-s − 9.12·43-s − 3.56·45-s + 3.68·47-s − 11.6·51-s + 3.12·53-s + 2.56·55-s − 2.87·57-s − 4·59-s + 9.36·61-s + 4.56·65-s + 6.24·67-s + ⋯
L(s)  = 1  − 1.47·3-s − 0.447·5-s + 1.18·9-s − 0.772·11-s − 1.26·13-s + 0.661·15-s + 1.10·17-s + 0.257·19-s + 1.06·23-s + 0.200·25-s − 0.276·27-s − 1.05·29-s + 1.14·33-s + 0.986·37-s + 1.87·39-s + 0.487·41-s − 1.39·43-s − 0.530·45-s + 0.537·47-s − 1.63·51-s + 0.428·53-s + 0.345·55-s − 0.381·57-s − 0.520·59-s + 1.19·61-s + 0.565·65-s + 0.763·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(31.3013\)
Root analytic conductor: \(5.59476\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3920} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3920,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
7 \( 1 \)
good3 \( 1 + 2.56T + 3T^{2} \)
11 \( 1 + 2.56T + 11T^{2} \)
13 \( 1 + 4.56T + 13T^{2} \)
17 \( 1 - 4.56T + 17T^{2} \)
19 \( 1 - 1.12T + 19T^{2} \)
23 \( 1 - 5.12T + 23T^{2} \)
29 \( 1 + 5.68T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 - 3.12T + 41T^{2} \)
43 \( 1 + 9.12T + 43T^{2} \)
47 \( 1 - 3.68T + 47T^{2} \)
53 \( 1 - 3.12T + 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 - 9.36T + 61T^{2} \)
67 \( 1 - 6.24T + 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + 4.24T + 73T^{2} \)
79 \( 1 - 6.56T + 79T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 + 7.12T + 89T^{2} \)
97 \( 1 - 14.8T + 97T^{2} \)
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

−7.74004540107140700552565840350, −7.41470083505652831045822138805, −6.60300811048339436766613681731, −5.67990010127077367454731080622, −5.17958190868467841924170862926, −4.62757011875040553518821123890, −3.49699879177161230680916513578, −2.48399588045007103902069471250, −1.02973534269655970867892478433, 0, 1.02973534269655970867892478433, 2.48399588045007103902069471250, 3.49699879177161230680916513578, 4.62757011875040553518821123890, 5.17958190868467841924170862926, 5.67990010127077367454731080622, 6.60300811048339436766613681731, 7.41470083505652831045822138805, 7.74004540107140700552565840350

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