Properties

Degree 2
Conductor $ 2^{2} \cdot 19 \cdot 79 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.98·5-s + 0.872·7-s − 3·9-s − 2.27·11-s − 2.48·13-s + 5.54·17-s + 19-s − 1.94·23-s + 10.9·25-s + 6.64·29-s + 5.65·31-s + 3.47·35-s − 6.90·37-s − 2.55·41-s + 3.83·43-s − 11.9·45-s − 3.63·47-s − 6.23·49-s + 7.39·53-s − 9.09·55-s + 2.04·59-s + 8.17·61-s − 2.61·63-s − 9.90·65-s + 5.55·67-s − 1.39·71-s + 2.85·73-s + ⋯
L(s)  = 1  + 1.78·5-s + 0.329·7-s − 9-s − 0.687·11-s − 0.688·13-s + 1.34·17-s + 0.229·19-s − 0.405·23-s + 2.18·25-s + 1.23·29-s + 1.01·31-s + 0.588·35-s − 1.13·37-s − 0.398·41-s + 0.584·43-s − 1.78·45-s − 0.530·47-s − 0.891·49-s + 1.01·53-s − 1.22·55-s + 0.266·59-s + 1.04·61-s − 0.329·63-s − 1.22·65-s + 0.679·67-s − 0.165·71-s + 0.334·73-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  =  0
Selberg data  =  $(2,\ 6004,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.747693439$
$L(\frac12)$  $\approx$  $2.747693439$
$L(\frac{3}{2})$   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 \( 1 + 3T^{2} \)
5 \( 1 - 3.98T + 5T^{2} \)
7 \( 1 - 0.872T + 7T^{2} \)
11 \( 1 + 2.27T + 11T^{2} \)
13 \( 1 + 2.48T + 13T^{2} \)
17 \( 1 - 5.54T + 17T^{2} \)
23 \( 1 + 1.94T + 23T^{2} \)
29 \( 1 - 6.64T + 29T^{2} \)
31 \( 1 - 5.65T + 31T^{2} \)
37 \( 1 + 6.90T + 37T^{2} \)
41 \( 1 + 2.55T + 41T^{2} \)
43 \( 1 - 3.83T + 43T^{2} \)
47 \( 1 + 3.63T + 47T^{2} \)
53 \( 1 - 7.39T + 53T^{2} \)
59 \( 1 - 2.04T + 59T^{2} \)
61 \( 1 - 8.17T + 61T^{2} \)
67 \( 1 - 5.55T + 67T^{2} \)
71 \( 1 + 1.39T + 71T^{2} \)
73 \( 1 - 2.85T + 73T^{2} \)
83 \( 1 + 0.801T + 83T^{2} \)
89 \( 1 - 8.75T + 89T^{2} \)
97 \( 1 - 0.744T + 97T^{2} \)
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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

−8.227872427476046781982910341682, −7.37055392637461183341176773322, −6.46143133709889967585339738940, −5.89179707125327586189481446081, −5.20170751292637443652693108032, −4.90083378066059502492191555192, −3.36214720557122211499005162021, −2.63707884384862693490496825964, −2.00776341032932074829068642931, −0.878574187835565482688778706510, 0.878574187835565482688778706510, 2.00776341032932074829068642931, 2.63707884384862693490496825964, 3.36214720557122211499005162021, 4.90083378066059502492191555192, 5.20170751292637443652693108032, 5.89179707125327586189481446081, 6.46143133709889967585339738940, 7.37055392637461183341176773322, 8.227872427476046781982910341682

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