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  − 1.48·3-s − 5-s − 3.53·7-s − 0.788·9-s − 0.193·11-s − 1.78·13-s + 1.48·15-s + 0.112·17-s − 1.44·19-s + 5.25·21-s + 7.37·23-s + 25-s + 5.63·27-s − 1.99·29-s − 2.75·31-s + 0.287·33-s + 3.53·35-s − 5.38·37-s + 2.65·39-s + 9.08·41-s + 7.95·43-s + 0.788·45-s − 9.63·47-s + 5.49·49-s − 0.167·51-s + 1.21·53-s + 0.193·55-s + ⋯
L(s)  = 1  − 0.858·3-s − 0.447·5-s − 1.33·7-s − 0.262·9-s − 0.0583·11-s − 0.495·13-s + 0.383·15-s + 0.0273·17-s − 0.331·19-s + 1.14·21-s + 1.53·23-s + 0.200·25-s + 1.08·27-s − 0.370·29-s − 0.495·31-s + 0.0500·33-s + 0.597·35-s − 0.885·37-s + 0.425·39-s + 1.41·41-s + 1.21·43-s + 0.117·45-s − 1.40·47-s + 0.784·49-s − 0.0234·51-s + 0.167·53-s + 0.0260·55-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 \( 1 + 1.48T + 3T^{2} \)
7 \( 1 + 3.53T + 7T^{2} \)
11 \( 1 + 0.193T + 11T^{2} \)
13 \( 1 + 1.78T + 13T^{2} \)
17 \( 1 - 0.112T + 17T^{2} \)
19 \( 1 + 1.44T + 19T^{2} \)
23 \( 1 - 7.37T + 23T^{2} \)
29 \( 1 + 1.99T + 29T^{2} \)
31 \( 1 + 2.75T + 31T^{2} \)
37 \( 1 + 5.38T + 37T^{2} \)
41 \( 1 - 9.08T + 41T^{2} \)
43 \( 1 - 7.95T + 43T^{2} \)
47 \( 1 + 9.63T + 47T^{2} \)
53 \( 1 - 1.21T + 53T^{2} \)
59 \( 1 + 4.20T + 59T^{2} \)
61 \( 1 - 6.14T + 61T^{2} \)
67 \( 1 - 8.98T + 67T^{2} \)
71 \( 1 + 7.09T + 71T^{2} \)
73 \( 1 - 11.1T + 73T^{2} \)
79 \( 1 - 5.03T + 79T^{2} \)
83 \( 1 - 13.2T + 83T^{2} \)
89 \( 1 - 16.2T + 89T^{2} \)
97 \( 1 + 6.48T + 97T^{2} \)
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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

−7.30388268792238752058923385570, −6.69341824459596485393208929047, −6.20333033288119274969627153191, −5.38424161883485390880190712808, −4.85633116557871954558573243522, −3.83816445295995713895867868344, −3.16206384680738968066625255628, −2.38717574778847880180105262598, −0.857228154443025368527627935911, 0, 0.857228154443025368527627935911, 2.38717574778847880180105262598, 3.16206384680738968066625255628, 3.83816445295995713895867868344, 4.85633116557871954558573243522, 5.38424161883485390880190712808, 6.20333033288119274969627153191, 6.69341824459596485393208929047, 7.30388268792238752058923385570

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