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  + 0.739·3-s + 2.08·5-s + 4.12·7-s − 2.45·9-s − 4.72·11-s − 3.93·13-s + 1.53·15-s + 1.15·17-s + 19-s + 3.04·21-s − 5.67·23-s − 0.661·25-s − 4.03·27-s − 8.45·29-s − 2.39·31-s − 3.49·33-s + 8.58·35-s + 5.34·37-s − 2.90·39-s + 0.419·41-s + 8.48·43-s − 5.11·45-s − 7.51·47-s + 9.98·49-s + 0.851·51-s − 6.91·53-s − 9.84·55-s + ⋯
L(s)  = 1  + 0.426·3-s + 0.931·5-s + 1.55·7-s − 0.817·9-s − 1.42·11-s − 1.09·13-s + 0.397·15-s + 0.279·17-s + 0.229·19-s + 0.664·21-s − 1.18·23-s − 0.132·25-s − 0.775·27-s − 1.56·29-s − 0.430·31-s − 0.608·33-s + 1.45·35-s + 0.878·37-s − 0.465·39-s + 0.0655·41-s + 1.29·43-s − 0.761·45-s − 1.09·47-s + 1.42·49-s + 0.119·51-s − 0.950·53-s − 1.32·55-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 \( 1 - 0.739T + 3T^{2} \)
5 \( 1 - 2.08T + 5T^{2} \)
7 \( 1 - 4.12T + 7T^{2} \)
11 \( 1 + 4.72T + 11T^{2} \)
13 \( 1 + 3.93T + 13T^{2} \)
17 \( 1 - 1.15T + 17T^{2} \)
23 \( 1 + 5.67T + 23T^{2} \)
29 \( 1 + 8.45T + 29T^{2} \)
31 \( 1 + 2.39T + 31T^{2} \)
37 \( 1 - 5.34T + 37T^{2} \)
41 \( 1 - 0.419T + 41T^{2} \)
43 \( 1 - 8.48T + 43T^{2} \)
47 \( 1 + 7.51T + 47T^{2} \)
53 \( 1 + 6.91T + 53T^{2} \)
59 \( 1 - 1.82T + 59T^{2} \)
61 \( 1 + 0.433T + 61T^{2} \)
67 \( 1 + 9.55T + 67T^{2} \)
71 \( 1 + 12.4T + 71T^{2} \)
73 \( 1 - 3.54T + 73T^{2} \)
83 \( 1 - 8.27T + 83T^{2} \)
89 \( 1 + 6.82T + 89T^{2} \)
97 \( 1 - 12.5T + 97T^{2} \)
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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

−7.85148341713440617251113930483, −7.38022103980318729482299754949, −5.99767263184767369501881696858, −5.52830371251347288954982680768, −5.03036414112675898836113341567, −4.15538785822327056925164514056, −2.92748650766118346903140588473, −2.24841594780617887954683263785, −1.70741108633124651966534340834, 0, 1.70741108633124651966534340834, 2.24841594780617887954683263785, 2.92748650766118346903140588473, 4.15538785822327056925164514056, 5.03036414112675898836113341567, 5.52830371251347288954982680768, 5.99767263184767369501881696858, 7.38022103980318729482299754949, 7.85148341713440617251113930483

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