Properties

Degree 2
Conductor $ 2^{2} \cdot 19 \cdot 79 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 1.57·3-s − 4.10·5-s − 3.99·7-s − 0.525·9-s − 5.59·11-s + 0.327·13-s + 6.45·15-s + 1.21·17-s + 19-s + 6.28·21-s + 0.458·23-s + 11.8·25-s + 5.54·27-s − 2.71·29-s + 0.456·31-s + 8.79·33-s + 16.4·35-s + 0.0670·37-s − 0.514·39-s − 1.54·41-s + 8.95·43-s + 2.15·45-s − 0.548·47-s + 8.97·49-s − 1.91·51-s − 5.96·53-s + 22.9·55-s + ⋯
L(s)  = 1  − 0.908·3-s − 1.83·5-s − 1.51·7-s − 0.175·9-s − 1.68·11-s + 0.0907·13-s + 1.66·15-s + 0.295·17-s + 0.229·19-s + 1.37·21-s + 0.0956·23-s + 2.36·25-s + 1.06·27-s − 0.503·29-s + 0.0819·31-s + 1.53·33-s + 2.77·35-s + 0.0110·37-s − 0.0823·39-s − 0.241·41-s + 1.36·43-s + 0.321·45-s − 0.0800·47-s + 1.28·49-s − 0.268·51-s − 0.819·53-s + 3.09·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6004\)    =    \(2^{2} \cdot 19 \cdot 79\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{6004} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 6004,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;19,\;79\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;19,\;79\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
19 \( 1 - T \)
79 \( 1 + T \)
good3 \( 1 + 1.57T + 3T^{2} \)
5 \( 1 + 4.10T + 5T^{2} \)
7 \( 1 + 3.99T + 7T^{2} \)
11 \( 1 + 5.59T + 11T^{2} \)
13 \( 1 - 0.327T + 13T^{2} \)
17 \( 1 - 1.21T + 17T^{2} \)
23 \( 1 - 0.458T + 23T^{2} \)
29 \( 1 + 2.71T + 29T^{2} \)
31 \( 1 - 0.456T + 31T^{2} \)
37 \( 1 - 0.0670T + 37T^{2} \)
41 \( 1 + 1.54T + 41T^{2} \)
43 \( 1 - 8.95T + 43T^{2} \)
47 \( 1 + 0.548T + 47T^{2} \)
53 \( 1 + 5.96T + 53T^{2} \)
59 \( 1 - 4.29T + 59T^{2} \)
61 \( 1 - 1.43T + 61T^{2} \)
67 \( 1 - 2.85T + 67T^{2} \)
71 \( 1 - 5.48T + 71T^{2} \)
73 \( 1 + 16.0T + 73T^{2} \)
83 \( 1 - 3.71T + 83T^{2} \)
89 \( 1 + 3.84T + 89T^{2} \)
97 \( 1 + 6.85T + 97T^{2} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.61256706030954481442535054219, −7.11189391249999223573382227221, −6.29847199381970044430036357365, −5.57255172962044953017317220482, −4.89963438122432598399717005885, −4.01664926379166001469881009471, −3.21957140877766600912776218144, −2.72253757289936569101546788667, −0.64776199887710157436986602223, 0, 0.64776199887710157436986602223, 2.72253757289936569101546788667, 3.21957140877766600912776218144, 4.01664926379166001469881009471, 4.89963438122432598399717005885, 5.57255172962044953017317220482, 6.29847199381970044430036357365, 7.11189391249999223573382227221, 7.61256706030954481442535054219

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