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  − 2.67·3-s + 3.55·5-s + 1.65·7-s + 4.14·9-s + 1.12·11-s − 2.27·13-s − 9.50·15-s − 3.07·17-s + 19-s − 4.43·21-s + 4.46·23-s + 7.62·25-s − 3.07·27-s − 3.82·29-s − 4.45·31-s − 2.99·33-s + 5.89·35-s − 11.5·37-s + 6.08·39-s − 9.46·41-s + 4.63·43-s + 14.7·45-s − 2.15·47-s − 4.25·49-s + 8.22·51-s − 7.35·53-s + 3.98·55-s + ⋯
L(s)  = 1  − 1.54·3-s + 1.58·5-s + 0.626·7-s + 1.38·9-s + 0.337·11-s − 0.631·13-s − 2.45·15-s − 0.746·17-s + 0.229·19-s − 0.967·21-s + 0.931·23-s + 1.52·25-s − 0.591·27-s − 0.710·29-s − 0.799·31-s − 0.521·33-s + 0.995·35-s − 1.90·37-s + 0.974·39-s − 1.47·41-s + 0.706·43-s + 2.19·45-s − 0.314·47-s − 0.607·49-s + 1.15·51-s − 1.01·53-s + 0.536·55-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 \( 1 + 2.67T + 3T^{2} \)
5 \( 1 - 3.55T + 5T^{2} \)
7 \( 1 - 1.65T + 7T^{2} \)
11 \( 1 - 1.12T + 11T^{2} \)
13 \( 1 + 2.27T + 13T^{2} \)
17 \( 1 + 3.07T + 17T^{2} \)
23 \( 1 - 4.46T + 23T^{2} \)
29 \( 1 + 3.82T + 29T^{2} \)
31 \( 1 + 4.45T + 31T^{2} \)
37 \( 1 + 11.5T + 37T^{2} \)
41 \( 1 + 9.46T + 41T^{2} \)
43 \( 1 - 4.63T + 43T^{2} \)
47 \( 1 + 2.15T + 47T^{2} \)
53 \( 1 + 7.35T + 53T^{2} \)
59 \( 1 + 3.21T + 59T^{2} \)
61 \( 1 - 11.6T + 61T^{2} \)
67 \( 1 + 13.1T + 67T^{2} \)
71 \( 1 + 12.2T + 71T^{2} \)
73 \( 1 + 13.5T + 73T^{2} \)
83 \( 1 + 10.9T + 83T^{2} \)
89 \( 1 + 4.05T + 89T^{2} \)
97 \( 1 - 16.8T + 97T^{2} \)
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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

−7.30829926880428598157118197613, −6.90587787950628697290735179008, −6.14197393301011575989894682437, −5.57432325191242305322496548694, −5.04087462230671361751656129528, −4.52474824974650785065179597024, −3.15105131440210899197791156282, −1.91154269231952280543856814494, −1.43228508078177691804410637923, 0, 1.43228508078177691804410637923, 1.91154269231952280543856814494, 3.15105131440210899197791156282, 4.52474824974650785065179597024, 5.04087462230671361751656129528, 5.57432325191242305322496548694, 6.14197393301011575989894682437, 6.90587787950628697290735179008, 7.30829926880428598157118197613

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