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  − 0.338·3-s − 3.08·5-s − 4.48·7-s − 2.88·9-s + 5.90·11-s + 1.46·13-s + 1.04·15-s − 2.53·17-s + 19-s + 1.51·21-s + 2.61·23-s + 4.51·25-s + 1.99·27-s + 2.21·29-s − 8.75·31-s − 2.00·33-s + 13.8·35-s − 1.37·37-s − 0.496·39-s + 9.51·41-s + 10.4·43-s + 8.89·45-s + 4.08·47-s + 13.1·49-s + 0.858·51-s − 0.486·53-s − 18.2·55-s + ⋯
L(s)  = 1  − 0.195·3-s − 1.37·5-s − 1.69·7-s − 0.961·9-s + 1.78·11-s + 0.406·13-s + 0.269·15-s − 0.614·17-s + 0.229·19-s + 0.331·21-s + 0.544·23-s + 0.903·25-s + 0.383·27-s + 0.411·29-s − 1.57·31-s − 0.348·33-s + 2.33·35-s − 0.226·37-s − 0.0794·39-s + 1.48·41-s + 1.58·43-s + 1.32·45-s + 0.595·47-s + 1.87·49-s + 0.120·51-s − 0.0668·53-s − 2.45·55-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 \( 1 + 0.338T + 3T^{2} \)
5 \( 1 + 3.08T + 5T^{2} \)
7 \( 1 + 4.48T + 7T^{2} \)
11 \( 1 - 5.90T + 11T^{2} \)
13 \( 1 - 1.46T + 13T^{2} \)
17 \( 1 + 2.53T + 17T^{2} \)
23 \( 1 - 2.61T + 23T^{2} \)
29 \( 1 - 2.21T + 29T^{2} \)
31 \( 1 + 8.75T + 31T^{2} \)
37 \( 1 + 1.37T + 37T^{2} \)
41 \( 1 - 9.51T + 41T^{2} \)
43 \( 1 - 10.4T + 43T^{2} \)
47 \( 1 - 4.08T + 47T^{2} \)
53 \( 1 + 0.486T + 53T^{2} \)
59 \( 1 + 11.1T + 59T^{2} \)
61 \( 1 - 2.62T + 61T^{2} \)
67 \( 1 + 10.4T + 67T^{2} \)
71 \( 1 + 3.34T + 71T^{2} \)
73 \( 1 - 10.8T + 73T^{2} \)
83 \( 1 - 5.71T + 83T^{2} \)
89 \( 1 + 12.0T + 89T^{2} \)
97 \( 1 - 10.8T + 97T^{2} \)
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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

−7.51548572060107142662585965824, −7.06997976731879243993369820101, −6.21023361274305024664509009865, −5.94443313968218510697067090388, −4.62308931428815919397014020807, −3.77136751859637407292006273943, −3.51804543349011129137633675637, −2.59081822083267405729614040418, −0.956738257191548000861116598894, 0, 0.956738257191548000861116598894, 2.59081822083267405729614040418, 3.51804543349011129137633675637, 3.77136751859637407292006273943, 4.62308931428815919397014020807, 5.94443313968218510697067090388, 6.21023361274305024664509009865, 7.06997976731879243993369820101, 7.51548572060107142662585965824

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