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  − 1.53·3-s − 5-s + 0.759·7-s − 0.646·9-s − 2.75·11-s − 4.86·13-s + 1.53·15-s + 0.830·17-s − 3.43·19-s − 1.16·21-s − 1.12·23-s + 25-s + 5.59·27-s + 9.75·29-s + 8.03·31-s + 4.23·33-s − 0.759·35-s + 5.57·37-s + 7.46·39-s − 4.04·41-s + 6.17·43-s + 0.646·45-s + 3.41·47-s − 6.42·49-s − 1.27·51-s + 5.80·53-s + 2.75·55-s + ⋯
L(s)  = 1  − 0.885·3-s − 0.447·5-s + 0.287·7-s − 0.215·9-s − 0.832·11-s − 1.34·13-s + 0.396·15-s + 0.201·17-s − 0.788·19-s − 0.254·21-s − 0.235·23-s + 0.200·25-s + 1.07·27-s + 1.81·29-s + 1.44·31-s + 0.737·33-s − 0.128·35-s + 0.916·37-s + 1.19·39-s − 0.631·41-s + 0.941·43-s + 0.0963·45-s + 0.497·47-s − 0.917·49-s − 0.178·51-s + 0.797·53-s + 0.372·55-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 \( 1 + 1.53T + 3T^{2} \)
7 \( 1 - 0.759T + 7T^{2} \)
11 \( 1 + 2.75T + 11T^{2} \)
13 \( 1 + 4.86T + 13T^{2} \)
17 \( 1 - 0.830T + 17T^{2} \)
19 \( 1 + 3.43T + 19T^{2} \)
23 \( 1 + 1.12T + 23T^{2} \)
29 \( 1 - 9.75T + 29T^{2} \)
31 \( 1 - 8.03T + 31T^{2} \)
37 \( 1 - 5.57T + 37T^{2} \)
41 \( 1 + 4.04T + 41T^{2} \)
43 \( 1 - 6.17T + 43T^{2} \)
47 \( 1 - 3.41T + 47T^{2} \)
53 \( 1 - 5.80T + 53T^{2} \)
59 \( 1 - 0.686T + 59T^{2} \)
61 \( 1 - 7.65T + 61T^{2} \)
67 \( 1 + 0.747T + 67T^{2} \)
71 \( 1 + 11.8T + 71T^{2} \)
73 \( 1 + 2.72T + 73T^{2} \)
79 \( 1 - 14.7T + 79T^{2} \)
83 \( 1 + 10.0T + 83T^{2} \)
89 \( 1 - 0.776T + 89T^{2} \)
97 \( 1 + 1.20T + 97T^{2} \)
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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

−7.50467188929080899914103874762, −6.69754139557230049702990040647, −6.13343943459043357688118067030, −5.27327056028766732866442639518, −4.78288724169914941713243272321, −4.20057063011138179652667396203, −2.86340443410318354227963524340, −2.44366924926037986842253476567, −0.937635391250608470685619627633, 0, 0.937635391250608470685619627633, 2.44366924926037986842253476567, 2.86340443410318354227963524340, 4.20057063011138179652667396203, 4.78288724169914941713243272321, 5.27327056028766732866442639518, 6.13343943459043357688118067030, 6.69754139557230049702990040647, 7.50467188929080899914103874762

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