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  + 2.40·3-s + 0.936·5-s − 3.55·7-s + 2.80·9-s − 1.35·11-s − 2.20·13-s + 2.25·15-s + 1.19·17-s + 19-s − 8.56·21-s − 5.26·23-s − 4.12·25-s − 0.478·27-s + 9.37·29-s + 0.532·31-s − 3.25·33-s − 3.33·35-s + 4.51·37-s − 5.30·39-s − 8.05·41-s + 12.1·43-s + 2.62·45-s − 0.711·47-s + 5.64·49-s + 2.87·51-s − 12.0·53-s − 1.26·55-s + ⋯
L(s)  = 1  + 1.39·3-s + 0.418·5-s − 1.34·7-s + 0.933·9-s − 0.407·11-s − 0.611·13-s + 0.582·15-s + 0.289·17-s + 0.229·19-s − 1.86·21-s − 1.09·23-s − 0.824·25-s − 0.0921·27-s + 1.74·29-s + 0.0956·31-s − 0.566·33-s − 0.562·35-s + 0.742·37-s − 0.850·39-s − 1.25·41-s + 1.84·43-s + 0.391·45-s − 0.103·47-s + 0.805·49-s + 0.402·51-s − 1.65·53-s − 0.170·55-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 \( 1 - 2.40T + 3T^{2} \)
5 \( 1 - 0.936T + 5T^{2} \)
7 \( 1 + 3.55T + 7T^{2} \)
11 \( 1 + 1.35T + 11T^{2} \)
13 \( 1 + 2.20T + 13T^{2} \)
17 \( 1 - 1.19T + 17T^{2} \)
23 \( 1 + 5.26T + 23T^{2} \)
29 \( 1 - 9.37T + 29T^{2} \)
31 \( 1 - 0.532T + 31T^{2} \)
37 \( 1 - 4.51T + 37T^{2} \)
41 \( 1 + 8.05T + 41T^{2} \)
43 \( 1 - 12.1T + 43T^{2} \)
47 \( 1 + 0.711T + 47T^{2} \)
53 \( 1 + 12.0T + 53T^{2} \)
59 \( 1 + 7.85T + 59T^{2} \)
61 \( 1 + 11.6T + 61T^{2} \)
67 \( 1 - 7.91T + 67T^{2} \)
71 \( 1 + 11.5T + 71T^{2} \)
73 \( 1 + 7.79T + 73T^{2} \)
83 \( 1 + 8.02T + 83T^{2} \)
89 \( 1 - 14.4T + 89T^{2} \)
97 \( 1 + 15.0T + 97T^{2} \)
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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

−7.87571427765272905439221679884, −7.16614947321715031026725427060, −6.29684971409721833457288076301, −5.78365590840919317549184436920, −4.63485142112025014695930450713, −3.84818828435509801297282838038, −2.90758470076528344980299201257, −2.70987242273639945767201789424, −1.60900310338533282701315999410, 0, 1.60900310338533282701315999410, 2.70987242273639945767201789424, 2.90758470076528344980299201257, 3.84818828435509801297282838038, 4.63485142112025014695930450713, 5.78365590840919317549184436920, 6.29684971409721833457288076301, 7.16614947321715031026725427060, 7.87571427765272905439221679884

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