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 − 2.70·3-s + 4-s + 5-s + 2.70·6-s − 1.21·7-s − 8-s + 4.29·9-s − 10-s − 5.34·11-s − 2.70·12-s + 4.99·13-s + 1.21·14-s − 2.70·15-s + 16-s + 1.52·17-s − 4.29·18-s − 5.54·19-s + 20-s + 3.27·21-s + 5.34·22-s − 2.17·23-s + 2.70·24-s + 25-s − 4.99·26-s − 3.49·27-s − 1.21·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.55·3-s + 0.5·4-s + 0.447·5-s + 1.10·6-s − 0.457·7-s − 0.353·8-s + 1.43·9-s − 0.316·10-s − 1.61·11-s − 0.779·12-s + 1.38·13-s + 0.323·14-s − 0.697·15-s + 0.250·16-s + 0.369·17-s − 1.01·18-s − 1.27·19-s + 0.223·20-s + 0.714·21-s + 1.13·22-s − 0.453·23-s + 0.551·24-s + 0.200·25-s − 0.979·26-s − 0.672·27-s − 0.228·28-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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 - T \)
401 \( 1 - T \)
good3 \( 1 + 2.70T + 3T^{2} \)
7 \( 1 + 1.21T + 7T^{2} \)
11 \( 1 + 5.34T + 11T^{2} \)
13 \( 1 - 4.99T + 13T^{2} \)
17 \( 1 - 1.52T + 17T^{2} \)
19 \( 1 + 5.54T + 19T^{2} \)
23 \( 1 + 2.17T + 23T^{2} \)
29 \( 1 - 5.67T + 29T^{2} \)
31 \( 1 - 7.09T + 31T^{2} \)
37 \( 1 - 1.20T + 37T^{2} \)
41 \( 1 + 10.3T + 41T^{2} \)
43 \( 1 + 12.2T + 43T^{2} \)
47 \( 1 - 12.4T + 47T^{2} \)
53 \( 1 - 10.8T + 53T^{2} \)
59 \( 1 - 3.79T + 59T^{2} \)
61 \( 1 + 6.62T + 61T^{2} \)
67 \( 1 + 7.49T + 67T^{2} \)
71 \( 1 + 0.842T + 71T^{2} \)
73 \( 1 - 8.60T + 73T^{2} \)
79 \( 1 - 4.99T + 79T^{2} \)
83 \( 1 + 1.39T + 83T^{2} \)
89 \( 1 - 1.31T + 89T^{2} \)
97 \( 1 + 9.84T + 97T^{2} \)
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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

−8.306076407224326578088793895469, −7.18131805348426792572176132301, −6.38279993580015570072060455852, −6.09086201336506549940768300490, −5.30929848578303671197409295493, −4.55321360679107665206907335551, −3.30294112490003428403205993015, −2.21741838233280892897559126398, −1.02625507922840265802455436450, 0, 1.02625507922840265802455436450, 2.21741838233280892897559126398, 3.30294112490003428403205993015, 4.55321360679107665206907335551, 5.30929848578303671197409295493, 6.09086201336506549940768300490, 6.38279993580015570072060455852, 7.18131805348426792572176132301, 8.306076407224326578088793895469

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