Properties

Degree 2
Conductor $ 2^{2} \cdot 5 \cdot 401 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.26·3-s − 5-s − 3.39·7-s + 2.10·9-s − 3.74·11-s − 4.09·13-s + 2.26·15-s + 1.81·17-s − 0.507·19-s + 7.66·21-s + 2.07·23-s + 25-s + 2.01·27-s − 2.08·29-s − 3.75·31-s + 8.46·33-s + 3.39·35-s + 5.62·37-s + 9.26·39-s − 1.53·41-s + 6.39·43-s − 2.10·45-s + 11.8·47-s + 4.50·49-s − 4.11·51-s − 11.1·53-s + 3.74·55-s + ⋯
L(s)  = 1  − 1.30·3-s − 0.447·5-s − 1.28·7-s + 0.702·9-s − 1.12·11-s − 1.13·13-s + 0.583·15-s + 0.441·17-s − 0.116·19-s + 1.67·21-s + 0.432·23-s + 0.200·25-s + 0.388·27-s − 0.386·29-s − 0.674·31-s + 1.47·33-s + 0.573·35-s + 0.925·37-s + 1.48·39-s − 0.240·41-s + 0.975·43-s − 0.314·45-s + 1.72·47-s + 0.643·49-s − 0.575·51-s − 1.53·53-s + 0.505·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8020\)    =    \(2^{2} \cdot 5 \cdot 401\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8020} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 8020,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;401\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;401\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
5 \( 1 + T \)
401 \( 1 - T \)
good3 \( 1 + 2.26T + 3T^{2} \)
7 \( 1 + 3.39T + 7T^{2} \)
11 \( 1 + 3.74T + 11T^{2} \)
13 \( 1 + 4.09T + 13T^{2} \)
17 \( 1 - 1.81T + 17T^{2} \)
19 \( 1 + 0.507T + 19T^{2} \)
23 \( 1 - 2.07T + 23T^{2} \)
29 \( 1 + 2.08T + 29T^{2} \)
31 \( 1 + 3.75T + 31T^{2} \)
37 \( 1 - 5.62T + 37T^{2} \)
41 \( 1 + 1.53T + 41T^{2} \)
43 \( 1 - 6.39T + 43T^{2} \)
47 \( 1 - 11.8T + 47T^{2} \)
53 \( 1 + 11.1T + 53T^{2} \)
59 \( 1 - 13.8T + 59T^{2} \)
61 \( 1 + 3.22T + 61T^{2} \)
67 \( 1 - 2.15T + 67T^{2} \)
71 \( 1 - 9.69T + 71T^{2} \)
73 \( 1 + 5.04T + 73T^{2} \)
79 \( 1 - 0.358T + 79T^{2} \)
83 \( 1 - 15.6T + 83T^{2} \)
89 \( 1 + 15.3T + 89T^{2} \)
97 \( 1 + 6.27T + 97T^{2} \)
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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

−7.35044221899181594861941143076, −6.77505758534341811974605545390, −6.01146540072154150533661781873, −5.44841248820438626075118568870, −4.88152318680249468657381887085, −4.00496766313765698748219386147, −3.06662445680496382000994169766, −2.37961986405156061258014887581, −0.74183695435515405195452736787, 0, 0.74183695435515405195452736787, 2.37961986405156061258014887581, 3.06662445680496382000994169766, 4.00496766313765698748219386147, 4.88152318680249468657381887085, 5.44841248820438626075118568870, 6.01146540072154150533661781873, 6.77505758534341811974605545390, 7.35044221899181594861941143076

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