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  + 1.67·3-s − 5-s − 2.58·7-s − 0.179·9-s + 4.59·11-s + 6.44·13-s − 1.67·15-s + 3.22·17-s − 7.28·19-s − 4.33·21-s − 7.91·23-s + 25-s − 5.33·27-s − 9.25·29-s − 3.08·31-s + 7.71·33-s + 2.58·35-s + 2.24·37-s + 10.8·39-s + 2.13·41-s + 7.12·43-s + 0.179·45-s − 1.12·47-s − 0.336·49-s + 5.41·51-s − 13.2·53-s − 4.59·55-s + ⋯
L(s)  = 1  + 0.969·3-s − 0.447·5-s − 0.975·7-s − 0.0597·9-s + 1.38·11-s + 1.78·13-s − 0.433·15-s + 0.781·17-s − 1.67·19-s − 0.946·21-s − 1.65·23-s + 0.200·25-s − 1.02·27-s − 1.71·29-s − 0.554·31-s + 1.34·33-s + 0.436·35-s + 0.368·37-s + 1.73·39-s + 0.333·41-s + 1.08·43-s + 0.0266·45-s − 0.163·47-s − 0.0481·49-s + 0.758·51-s − 1.81·53-s − 0.619·55-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 \( 1 - 1.67T + 3T^{2} \)
7 \( 1 + 2.58T + 7T^{2} \)
11 \( 1 - 4.59T + 11T^{2} \)
13 \( 1 - 6.44T + 13T^{2} \)
17 \( 1 - 3.22T + 17T^{2} \)
19 \( 1 + 7.28T + 19T^{2} \)
23 \( 1 + 7.91T + 23T^{2} \)
29 \( 1 + 9.25T + 29T^{2} \)
31 \( 1 + 3.08T + 31T^{2} \)
37 \( 1 - 2.24T + 37T^{2} \)
41 \( 1 - 2.13T + 41T^{2} \)
43 \( 1 - 7.12T + 43T^{2} \)
47 \( 1 + 1.12T + 47T^{2} \)
53 \( 1 + 13.2T + 53T^{2} \)
59 \( 1 - 3.97T + 59T^{2} \)
61 \( 1 - 8.25T + 61T^{2} \)
67 \( 1 - 0.637T + 67T^{2} \)
71 \( 1 + 3.08T + 71T^{2} \)
73 \( 1 + 5.63T + 73T^{2} \)
79 \( 1 - 8.87T + 79T^{2} \)
83 \( 1 - 12.7T + 83T^{2} \)
89 \( 1 - 5.95T + 89T^{2} \)
97 \( 1 + 15.2T + 97T^{2} \)
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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

−7.75812677535637324870493712593, −6.66544668622225753485101492104, −6.20350170640871369889089822618, −5.68599224558122260186169128537, −4.08364923274974300038630596623, −3.84064930265631564879871982032, −3.37591707878431819354060233816, −2.27019925582061829208394311903, −1.41402069269485335909508544716, 0, 1.41402069269485335909508544716, 2.27019925582061829208394311903, 3.37591707878431819354060233816, 3.84064930265631564879871982032, 4.08364923274974300038630596623, 5.68599224558122260186169128537, 6.20350170640871369889089822618, 6.66544668622225753485101492104, 7.75812677535637324870493712593

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