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  + 0.267·3-s + 3.98·5-s − 3.08·7-s − 2.92·9-s + 5.66·11-s − 4.10·13-s + 1.06·15-s + 0.737·17-s + 19-s − 0.825·21-s − 6.58·23-s + 10.8·25-s − 1.58·27-s − 7.16·29-s − 7.97·31-s + 1.51·33-s − 12.2·35-s + 4.21·37-s − 1.09·39-s − 10.2·41-s − 0.0742·43-s − 11.6·45-s − 6.74·47-s + 2.52·49-s + 0.197·51-s + 6.32·53-s + 22.5·55-s + ⋯
L(s)  = 1  + 0.154·3-s + 1.78·5-s − 1.16·7-s − 0.976·9-s + 1.70·11-s − 1.13·13-s + 0.275·15-s + 0.178·17-s + 0.229·19-s − 0.180·21-s − 1.37·23-s + 2.17·25-s − 0.305·27-s − 1.33·29-s − 1.43·31-s + 0.263·33-s − 2.07·35-s + 0.692·37-s − 0.175·39-s − 1.59·41-s − 0.0113·43-s − 1.73·45-s − 0.984·47-s + 0.360·49-s + 0.0276·51-s + 0.869·53-s + 3.04·55-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 \( 1 - 0.267T + 3T^{2} \)
5 \( 1 - 3.98T + 5T^{2} \)
7 \( 1 + 3.08T + 7T^{2} \)
11 \( 1 - 5.66T + 11T^{2} \)
13 \( 1 + 4.10T + 13T^{2} \)
17 \( 1 - 0.737T + 17T^{2} \)
23 \( 1 + 6.58T + 23T^{2} \)
29 \( 1 + 7.16T + 29T^{2} \)
31 \( 1 + 7.97T + 31T^{2} \)
37 \( 1 - 4.21T + 37T^{2} \)
41 \( 1 + 10.2T + 41T^{2} \)
43 \( 1 + 0.0742T + 43T^{2} \)
47 \( 1 + 6.74T + 47T^{2} \)
53 \( 1 - 6.32T + 53T^{2} \)
59 \( 1 - 4.35T + 59T^{2} \)
61 \( 1 + 15.4T + 61T^{2} \)
67 \( 1 - 1.03T + 67T^{2} \)
71 \( 1 + 2.60T + 71T^{2} \)
73 \( 1 + 1.11T + 73T^{2} \)
83 \( 1 - 17.0T + 83T^{2} \)
89 \( 1 + 5.80T + 89T^{2} \)
97 \( 1 + 8.59T + 97T^{2} \)
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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

−7.61782559538557038750973307806, −6.76892262631367586013094696663, −6.24986721758983371044229466532, −5.75852379570840015680962778733, −5.08536557645571392517423123429, −3.84847129258556384409718597278, −3.16350636800569143360928292818, −2.24152150414128336423314447630, −1.59352188894863065929323614885, 0, 1.59352188894863065929323614885, 2.24152150414128336423314447630, 3.16350636800569143360928292818, 3.84847129258556384409718597278, 5.08536557645571392517423123429, 5.75852379570840015680962778733, 6.24986721758983371044229466532, 6.76892262631367586013094696663, 7.61782559538557038750973307806

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