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  + 1.41·3-s − 0.774·5-s − 1.47·7-s − 1.00·9-s + 1.39·11-s − 0.835·13-s − 1.09·15-s + 6.36·17-s + 19-s − 2.08·21-s − 2.70·23-s − 4.40·25-s − 5.65·27-s − 6.34·29-s + 8.44·31-s + 1.97·33-s + 1.13·35-s − 5.59·37-s − 1.18·39-s + 8.34·41-s + 2.34·43-s + 0.775·45-s − 13.0·47-s − 4.83·49-s + 8.99·51-s + 6.78·53-s − 1.08·55-s + ⋯
L(s)  = 1  + 0.816·3-s − 0.346·5-s − 0.556·7-s − 0.333·9-s + 0.421·11-s − 0.231·13-s − 0.282·15-s + 1.54·17-s + 0.229·19-s − 0.454·21-s − 0.563·23-s − 0.880·25-s − 1.08·27-s − 1.17·29-s + 1.51·31-s + 0.344·33-s + 0.192·35-s − 0.920·37-s − 0.189·39-s + 1.30·41-s + 0.358·43-s + 0.115·45-s − 1.89·47-s − 0.690·49-s + 1.25·51-s + 0.932·53-s − 0.146·55-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 \( 1 - 1.41T + 3T^{2} \)
5 \( 1 + 0.774T + 5T^{2} \)
7 \( 1 + 1.47T + 7T^{2} \)
11 \( 1 - 1.39T + 11T^{2} \)
13 \( 1 + 0.835T + 13T^{2} \)
17 \( 1 - 6.36T + 17T^{2} \)
23 \( 1 + 2.70T + 23T^{2} \)
29 \( 1 + 6.34T + 29T^{2} \)
31 \( 1 - 8.44T + 31T^{2} \)
37 \( 1 + 5.59T + 37T^{2} \)
41 \( 1 - 8.34T + 41T^{2} \)
43 \( 1 - 2.34T + 43T^{2} \)
47 \( 1 + 13.0T + 47T^{2} \)
53 \( 1 - 6.78T + 53T^{2} \)
59 \( 1 + 9.23T + 59T^{2} \)
61 \( 1 + 5.97T + 61T^{2} \)
67 \( 1 + 0.612T + 67T^{2} \)
71 \( 1 + 2.89T + 71T^{2} \)
73 \( 1 - 3.61T + 73T^{2} \)
83 \( 1 + 9.51T + 83T^{2} \)
89 \( 1 - 1.80T + 89T^{2} \)
97 \( 1 + 11.7T + 97T^{2} \)
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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

−7.79787184168698101558707635872, −7.27685869458823217137239402114, −6.19585685702268036727662049053, −5.75049594166627474771852425511, −4.73432524032396452936243608680, −3.72484002549577564804954459871, −3.33360861575393565917495973627, −2.49657352781182951751658762089, −1.42103566000731217464926442850, 0, 1.42103566000731217464926442850, 2.49657352781182951751658762089, 3.33360861575393565917495973627, 3.72484002549577564804954459871, 4.73432524032396452936243608680, 5.75049594166627474771852425511, 6.19585685702268036727662049053, 7.27685869458823217137239402114, 7.79787184168698101558707635872

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