Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.48·3-s + 1.00·5-s + 0.658·7-s + 3.16·9-s − 5.12·11-s − 2.19·13-s + 2.50·15-s − 4.93·17-s + 19-s + 1.63·21-s + 2.25·23-s − 3.98·25-s + 0.403·27-s − 3.27·29-s − 6.68·31-s − 12.7·33-s + 0.664·35-s + 4.75·37-s − 5.43·39-s − 6.29·41-s − 7.36·43-s + 3.19·45-s − 8.45·47-s − 6.56·49-s − 12.2·51-s + 6.21·53-s − 5.17·55-s + ⋯
L(s)  = 1  + 1.43·3-s + 0.451·5-s + 0.248·7-s + 1.05·9-s − 1.54·11-s − 0.607·13-s + 0.646·15-s − 1.19·17-s + 0.229·19-s + 0.356·21-s + 0.469·23-s − 0.796·25-s + 0.0776·27-s − 0.608·29-s − 1.20·31-s − 2.21·33-s + 0.112·35-s + 0.781·37-s − 0.870·39-s − 0.983·41-s − 1.12·43-s + 0.475·45-s − 1.23·47-s − 0.938·49-s − 1.71·51-s + 0.853·53-s − 0.697·55-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 \( 1 - 2.48T + 3T^{2} \)
5 \( 1 - 1.00T + 5T^{2} \)
7 \( 1 - 0.658T + 7T^{2} \)
11 \( 1 + 5.12T + 11T^{2} \)
13 \( 1 + 2.19T + 13T^{2} \)
17 \( 1 + 4.93T + 17T^{2} \)
23 \( 1 - 2.25T + 23T^{2} \)
29 \( 1 + 3.27T + 29T^{2} \)
31 \( 1 + 6.68T + 31T^{2} \)
37 \( 1 - 4.75T + 37T^{2} \)
41 \( 1 + 6.29T + 41T^{2} \)
43 \( 1 + 7.36T + 43T^{2} \)
47 \( 1 + 8.45T + 47T^{2} \)
53 \( 1 - 6.21T + 53T^{2} \)
59 \( 1 - 2.76T + 59T^{2} \)
61 \( 1 - 1.82T + 61T^{2} \)
67 \( 1 - 7.94T + 67T^{2} \)
71 \( 1 - 13.7T + 71T^{2} \)
73 \( 1 + 3.62T + 73T^{2} \)
83 \( 1 + 12.7T + 83T^{2} \)
89 \( 1 - 10.3T + 89T^{2} \)
97 \( 1 - 8.58T + 97T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.78614465283573301758761031123, −7.31378119464755716667948325583, −6.44408503117723342050137312764, −5.35551473171815897735114299153, −4.90941014308574698461909747886, −3.84759552254540565072523535097, −3.07812707033569525885183708608, −2.28073760498055780276754522943, −1.86181145829630040651669316719, 0, 1.86181145829630040651669316719, 2.28073760498055780276754522943, 3.07812707033569525885183708608, 3.84759552254540565072523535097, 4.90941014308574698461909747886, 5.35551473171815897735114299153, 6.44408503117723342050137312764, 7.31378119464755716667948325583, 7.78614465283573301758761031123

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