Properties

Label 2-7616-1.1-c1-0-175
Degree $2$
Conductor $7616$
Sign $-1$
Analytic cond. $60.8140$
Root an. cond. $7.79833$
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  + 1.30·3-s + 2.30·5-s + 7-s − 1.30·9-s − 6.60·13-s + 3·15-s + 17-s − 6.60·19-s + 1.30·21-s + 0.302·25-s − 5.60·27-s + 4.30·31-s + 2.30·35-s + 2.60·37-s − 8.60·39-s + 3.90·41-s − 7.30·43-s − 3.00·45-s + 4.60·47-s + 49-s + 1.30·51-s + 3.69·53-s − 8.60·57-s − 9.21·59-s + 7.90·61-s − 1.30·63-s − 15.2·65-s + ⋯
L(s)  = 1  + 0.752·3-s + 1.02·5-s + 0.377·7-s − 0.434·9-s − 1.83·13-s + 0.774·15-s + 0.242·17-s − 1.51·19-s + 0.284·21-s + 0.0605·25-s − 1.07·27-s + 0.772·31-s + 0.389·35-s + 0.428·37-s − 1.37·39-s + 0.610·41-s − 1.11·43-s − 0.447·45-s + 0.671·47-s + 0.142·49-s + 0.182·51-s + 0.507·53-s − 1.13·57-s − 1.19·59-s + 1.01·61-s − 0.164·63-s − 1.88·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7616\)    =    \(2^{6} \cdot 7 \cdot 17\)
Sign: $-1$
Analytic conductor: \(60.8140\)
Root analytic conductor: \(7.79833\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7616,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 - T \)
17 \( 1 - T \)
good3 \( 1 - 1.30T + 3T^{2} \)
5 \( 1 - 2.30T + 5T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 6.60T + 13T^{2} \)
19 \( 1 + 6.60T + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 4.30T + 31T^{2} \)
37 \( 1 - 2.60T + 37T^{2} \)
41 \( 1 - 3.90T + 41T^{2} \)
43 \( 1 + 7.30T + 43T^{2} \)
47 \( 1 - 4.60T + 47T^{2} \)
53 \( 1 - 3.69T + 53T^{2} \)
59 \( 1 + 9.21T + 59T^{2} \)
61 \( 1 - 7.90T + 61T^{2} \)
67 \( 1 - 1.69T + 67T^{2} \)
71 \( 1 + 7.81T + 71T^{2} \)
73 \( 1 + 7.90T + 73T^{2} \)
79 \( 1 - 12.6T + 79T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + 16.6T + 89T^{2} \)
97 \( 1 + 6.30T + 97T^{2} \)
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

−7.63016014022101103487022071276, −6.88504614651688014263683805649, −6.09004899815329670739141928375, −5.44876031132231184750358307196, −4.71567236984094831812065547465, −3.95725226192777045142416743743, −2.67231087050597624240718026027, −2.48137675798750353652743958826, −1.59414313682064723442830397276, 0, 1.59414313682064723442830397276, 2.48137675798750353652743958826, 2.67231087050597624240718026027, 3.95725226192777045142416743743, 4.71567236984094831812065547465, 5.44876031132231184750358307196, 6.09004899815329670739141928375, 6.88504614651688014263683805649, 7.63016014022101103487022071276

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