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 + 1.23·3-s + 4-s − 5-s − 1.23·6-s + 3.50·7-s − 8-s − 1.48·9-s + 10-s − 1.86·11-s + 1.23·12-s − 6.67·13-s − 3.50·14-s − 1.23·15-s + 16-s + 6.33·17-s + 1.48·18-s + 4.59·19-s − 20-s + 4.31·21-s + 1.86·22-s − 8.71·23-s − 1.23·24-s + 25-s + 6.67·26-s − 5.51·27-s + 3.50·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.710·3-s + 0.5·4-s − 0.447·5-s − 0.502·6-s + 1.32·7-s − 0.353·8-s − 0.495·9-s + 0.316·10-s − 0.562·11-s + 0.355·12-s − 1.85·13-s − 0.936·14-s − 0.317·15-s + 0.250·16-s + 1.53·17-s + 0.350·18-s + 1.05·19-s − 0.223·20-s + 0.941·21-s + 0.397·22-s − 1.81·23-s − 0.251·24-s + 0.200·25-s + 1.30·26-s − 1.06·27-s + 0.662·28-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 \( 1 - 1.23T + 3T^{2} \)
7 \( 1 - 3.50T + 7T^{2} \)
11 \( 1 + 1.86T + 11T^{2} \)
13 \( 1 + 6.67T + 13T^{2} \)
17 \( 1 - 6.33T + 17T^{2} \)
19 \( 1 - 4.59T + 19T^{2} \)
23 \( 1 + 8.71T + 23T^{2} \)
29 \( 1 - 9.04T + 29T^{2} \)
31 \( 1 + 8.47T + 31T^{2} \)
37 \( 1 + 3.07T + 37T^{2} \)
41 \( 1 + 3.18T + 41T^{2} \)
43 \( 1 + 2.83T + 43T^{2} \)
47 \( 1 - 8.62T + 47T^{2} \)
53 \( 1 - 6.47T + 53T^{2} \)
59 \( 1 + 5.42T + 59T^{2} \)
61 \( 1 + 4.38T + 61T^{2} \)
67 \( 1 - 2.27T + 67T^{2} \)
71 \( 1 + 0.782T + 71T^{2} \)
73 \( 1 + 1.86T + 73T^{2} \)
79 \( 1 + 8.08T + 79T^{2} \)
83 \( 1 + 15.8T + 83T^{2} \)
89 \( 1 + 11.8T + 89T^{2} \)
97 \( 1 + 15.0T + 97T^{2} \)
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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

−8.085822830331745043823206951676, −7.59538383752524923666113944645, −7.16490903446515738027383979191, −5.62350141335110863420245195331, −5.25409019838056900280280519343, −4.20672799512062730155830827841, −3.11421463175290000119842797568, −2.44020690715435400912951400490, −1.48455686474206213660930715879, 0, 1.48455686474206213660930715879, 2.44020690715435400912951400490, 3.11421463175290000119842797568, 4.20672799512062730155830827841, 5.25409019838056900280280519343, 5.62350141335110863420245195331, 7.16490903446515738027383979191, 7.59538383752524923666113944645, 8.085822830331745043823206951676

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