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  − 1.59·3-s + 2.11·5-s + 0.323·7-s − 0.457·9-s − 2.74·11-s − 4.84·13-s − 3.37·15-s + 4.66·17-s + 19-s − 0.515·21-s + 1.81·23-s − 0.515·25-s + 5.51·27-s − 3.81·29-s + 3.28·31-s + 4.37·33-s + 0.684·35-s + 7.27·37-s + 7.72·39-s + 3.37·41-s − 7.47·43-s − 0.967·45-s + 10.2·47-s − 6.89·49-s − 7.43·51-s − 1.33·53-s − 5.81·55-s + ⋯
L(s)  = 1  − 0.920·3-s + 0.947·5-s + 0.122·7-s − 0.152·9-s − 0.828·11-s − 1.34·13-s − 0.871·15-s + 1.13·17-s + 0.229·19-s − 0.112·21-s + 0.377·23-s − 0.103·25-s + 1.06·27-s − 0.707·29-s + 0.590·31-s + 0.762·33-s + 0.115·35-s + 1.19·37-s + 1.23·39-s + 0.526·41-s − 1.14·43-s − 0.144·45-s + 1.49·47-s − 0.985·49-s − 1.04·51-s − 0.182·53-s − 0.784·55-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 \( 1 + 1.59T + 3T^{2} \)
5 \( 1 - 2.11T + 5T^{2} \)
7 \( 1 - 0.323T + 7T^{2} \)
11 \( 1 + 2.74T + 11T^{2} \)
13 \( 1 + 4.84T + 13T^{2} \)
17 \( 1 - 4.66T + 17T^{2} \)
23 \( 1 - 1.81T + 23T^{2} \)
29 \( 1 + 3.81T + 29T^{2} \)
31 \( 1 - 3.28T + 31T^{2} \)
37 \( 1 - 7.27T + 37T^{2} \)
41 \( 1 - 3.37T + 41T^{2} \)
43 \( 1 + 7.47T + 43T^{2} \)
47 \( 1 - 10.2T + 47T^{2} \)
53 \( 1 + 1.33T + 53T^{2} \)
59 \( 1 - 5.58T + 59T^{2} \)
61 \( 1 - 2.74T + 61T^{2} \)
67 \( 1 - 5.70T + 67T^{2} \)
71 \( 1 + 0.576T + 71T^{2} \)
73 \( 1 + 12.9T + 73T^{2} \)
83 \( 1 + 9.16T + 83T^{2} \)
89 \( 1 + 5.29T + 89T^{2} \)
97 \( 1 + 13.8T + 97T^{2} \)
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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

−7.65833063639738885074090377099, −6.95684550962151102258263287847, −6.10975068146688419891600751529, −5.42271534878141131969316529329, −5.25068103208726375639070981906, −4.29133501313412926771524318825, −2.96671573328564682029309008644, −2.39770786283067941414589176733, −1.21016927207050466288531287407, 0, 1.21016927207050466288531287407, 2.39770786283067941414589176733, 2.96671573328564682029309008644, 4.29133501313412926771524318825, 5.25068103208726375639070981906, 5.42271534878141131969316529329, 6.10975068146688419891600751529, 6.95684550962151102258263287847, 7.65833063639738885074090377099

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