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 − 1.25·3-s + 4-s + 5-s + 1.25·6-s + 1.88·7-s − 8-s − 1.41·9-s − 10-s − 3.48·11-s − 1.25·12-s − 6.95·13-s − 1.88·14-s − 1.25·15-s + 16-s + 2.07·17-s + 1.41·18-s + 3.03·19-s + 20-s − 2.37·21-s + 3.48·22-s + 6.09·23-s + 1.25·24-s + 25-s + 6.95·26-s + 5.55·27-s + 1.88·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.726·3-s + 0.5·4-s + 0.447·5-s + 0.513·6-s + 0.713·7-s − 0.353·8-s − 0.471·9-s − 0.316·10-s − 1.05·11-s − 0.363·12-s − 1.92·13-s − 0.504·14-s − 0.324·15-s + 0.250·16-s + 0.504·17-s + 0.333·18-s + 0.696·19-s + 0.223·20-s − 0.518·21-s + 0.743·22-s + 1.27·23-s + 0.256·24-s + 0.200·25-s + 1.36·26-s + 1.06·27-s + 0.356·28-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 \( 1 + 1.25T + 3T^{2} \)
7 \( 1 - 1.88T + 7T^{2} \)
11 \( 1 + 3.48T + 11T^{2} \)
13 \( 1 + 6.95T + 13T^{2} \)
17 \( 1 - 2.07T + 17T^{2} \)
19 \( 1 - 3.03T + 19T^{2} \)
23 \( 1 - 6.09T + 23T^{2} \)
29 \( 1 + 1.90T + 29T^{2} \)
31 \( 1 - 8.71T + 31T^{2} \)
37 \( 1 + 7.53T + 37T^{2} \)
41 \( 1 - 2.87T + 41T^{2} \)
43 \( 1 - 8.26T + 43T^{2} \)
47 \( 1 + 0.387T + 47T^{2} \)
53 \( 1 - 10.1T + 53T^{2} \)
59 \( 1 + 11.5T + 59T^{2} \)
61 \( 1 + 6.34T + 61T^{2} \)
67 \( 1 + 14.2T + 67T^{2} \)
71 \( 1 + 8.46T + 71T^{2} \)
73 \( 1 - 7.64T + 73T^{2} \)
79 \( 1 - 7.07T + 79T^{2} \)
83 \( 1 - 4.66T + 83T^{2} \)
89 \( 1 - 1.33T + 89T^{2} \)
97 \( 1 + 3.66T + 97T^{2} \)
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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

−7.896103193234611321490772479364, −7.53183385913984851296504934511, −6.72953518168572452345191178947, −5.74532347832794512945696144455, −5.15851174181380552301826152910, −4.71856471765542492231992833624, −2.95551592826889692691714255862, −2.47730620088101945932137289346, −1.18638041785933017448880353192, 0, 1.18638041785933017448880353192, 2.47730620088101945932137289346, 2.95551592826889692691714255862, 4.71856471765542492231992833624, 5.15851174181380552301826152910, 5.74532347832794512945696144455, 6.72953518168572452345191178947, 7.53183385913984851296504934511, 7.896103193234611321490772479364

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