Properties

Degree 2
Conductor $ 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-s − 1.21·3-s + 4-s − 5-s + 1.21·6-s + 0.441·7-s − 8-s − 1.53·9-s + 10-s + 0.781·11-s − 1.21·12-s − 4.67·13-s − 0.441·14-s + 1.21·15-s + 16-s + 0.347·17-s + 1.53·18-s + 0.947·19-s − 20-s − 0.535·21-s − 0.781·22-s − 2.26·23-s + 1.21·24-s + 25-s + 4.67·26-s + 5.49·27-s + 0.441·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.699·3-s + 0.5·4-s − 0.447·5-s + 0.494·6-s + 0.166·7-s − 0.353·8-s − 0.510·9-s + 0.316·10-s + 0.235·11-s − 0.349·12-s − 1.29·13-s − 0.118·14-s + 0.312·15-s + 0.250·16-s + 0.0842·17-s + 0.360·18-s + 0.217·19-s − 0.223·20-s − 0.116·21-s − 0.166·22-s − 0.472·23-s + 0.247·24-s + 0.200·25-s + 0.917·26-s + 1.05·27-s + 0.0834·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4010\)    =    \(2 \cdot 5 \cdot 401\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{4010} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 4010,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 + T \)
5 \( 1 + T \)
401 \( 1 + T \)
good3 \( 1 + 1.21T + 3T^{2} \)
7 \( 1 - 0.441T + 7T^{2} \)
11 \( 1 - 0.781T + 11T^{2} \)
13 \( 1 + 4.67T + 13T^{2} \)
17 \( 1 - 0.347T + 17T^{2} \)
19 \( 1 - 0.947T + 19T^{2} \)
23 \( 1 + 2.26T + 23T^{2} \)
29 \( 1 - 8.38T + 29T^{2} \)
31 \( 1 - 7.83T + 31T^{2} \)
37 \( 1 - 7.03T + 37T^{2} \)
41 \( 1 + 4.97T + 41T^{2} \)
43 \( 1 - 1.53T + 43T^{2} \)
47 \( 1 - 2.09T + 47T^{2} \)
53 \( 1 + 9.21T + 53T^{2} \)
59 \( 1 - 7.20T + 59T^{2} \)
61 \( 1 + 13.0T + 61T^{2} \)
67 \( 1 + 11.2T + 67T^{2} \)
71 \( 1 - 9.64T + 71T^{2} \)
73 \( 1 + 5.47T + 73T^{2} \)
79 \( 1 + 1.92T + 79T^{2} \)
83 \( 1 - 8.36T + 83T^{2} \)
89 \( 1 - 7.27T + 89T^{2} \)
97 \( 1 - 11.6T + 97T^{2} \)
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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

−8.045533610488998238142213431031, −7.51162835869472312073400172410, −6.53844693463555279023383155803, −6.12717093610571066232062788788, −4.99845170560557356198967057118, −4.54166235374781992509736239443, −3.17864461228497707040937586598, −2.44441774550427368580940787900, −1.06694878469586337632051625557, 0, 1.06694878469586337632051625557, 2.44441774550427368580940787900, 3.17864461228497707040937586598, 4.54166235374781992509736239443, 4.99845170560557356198967057118, 6.12717093610571066232062788788, 6.53844693463555279023383155803, 7.51162835869472312073400172410, 8.045533610488998238142213431031

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