Properties

Label 2-7600-1.1-c1-0-152
Degree $2$
Conductor $7600$
Sign $-1$
Analytic cond. $60.6863$
Root an. cond. $7.79014$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.27·3-s + 0.726·7-s − 1.37·9-s + 0.273·11-s + 5.95·13-s − 5.27·17-s − 19-s + 0.924·21-s − 3.67·23-s − 5.57·27-s − 2.27·29-s − 3.19·31-s + 0.348·33-s − 8.12·37-s + 7.58·39-s − 9.43·41-s + 9.81·43-s + 12.1·47-s − 6.47·49-s − 6.71·51-s − 5.69·53-s − 1.27·57-s + 4.20·59-s − 0.103·61-s − 0.999·63-s − 11.7·67-s − 4.68·69-s + ⋯
L(s)  = 1  + 0.735·3-s + 0.274·7-s − 0.459·9-s + 0.0825·11-s + 1.65·13-s − 1.27·17-s − 0.229·19-s + 0.201·21-s − 0.767·23-s − 1.07·27-s − 0.422·29-s − 0.574·31-s + 0.0607·33-s − 1.33·37-s + 1.21·39-s − 1.47·41-s + 1.49·43-s + 1.77·47-s − 0.924·49-s − 0.940·51-s − 0.782·53-s − 0.168·57-s + 0.547·59-s − 0.0132·61-s − 0.125·63-s − 1.43·67-s − 0.564·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7600\)    =    \(2^{4} \cdot 5^{2} \cdot 19\)
Sign: $-1$
Analytic conductor: \(60.6863\)
Root analytic conductor: \(7.79014\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7600,\ (\ :1/2),\ -1)\)

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 \( 1 \)
19 \( 1 + T \)
good3 \( 1 - 1.27T + 3T^{2} \)
7 \( 1 - 0.726T + 7T^{2} \)
11 \( 1 - 0.273T + 11T^{2} \)
13 \( 1 - 5.95T + 13T^{2} \)
17 \( 1 + 5.27T + 17T^{2} \)
23 \( 1 + 3.67T + 23T^{2} \)
29 \( 1 + 2.27T + 29T^{2} \)
31 \( 1 + 3.19T + 31T^{2} \)
37 \( 1 + 8.12T + 37T^{2} \)
41 \( 1 + 9.43T + 41T^{2} \)
43 \( 1 - 9.81T + 43T^{2} \)
47 \( 1 - 12.1T + 47T^{2} \)
53 \( 1 + 5.69T + 53T^{2} \)
59 \( 1 - 4.20T + 59T^{2} \)
61 \( 1 + 0.103T + 61T^{2} \)
67 \( 1 + 11.7T + 67T^{2} \)
71 \( 1 + 5.75T + 71T^{2} \)
73 \( 1 + 6.67T + 73T^{2} \)
79 \( 1 + 3.87T + 79T^{2} \)
83 \( 1 + 0.488T + 83T^{2} \)
89 \( 1 + 16.4T + 89T^{2} \)
97 \( 1 - 4.44T + 97T^{2} \)
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

−7.64530390296043936598963081622, −6.89244690367551691333498493110, −6.06403308641084553298425315647, −5.61000459636287331195023958020, −4.49482366776071365596298165351, −3.84035847487999676192514607206, −3.19106099124531897616364711897, −2.20923838849575574211228593640, −1.51671879359981659705205137181, 0, 1.51671879359981659705205137181, 2.20923838849575574211228593640, 3.19106099124531897616364711897, 3.84035847487999676192514607206, 4.49482366776071365596298165351, 5.61000459636287331195023958020, 6.06403308641084553298425315647, 6.89244690367551691333498493110, 7.64530390296043936598963081622

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