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 + 2·3-s + 4-s − 5-s − 2·6-s − 8-s + 9-s + 10-s + 3·11-s + 2·12-s + 13-s − 2·15-s + 16-s − 8·17-s − 18-s − 20-s − 3·22-s − 2·24-s + 25-s − 26-s − 4·27-s − 4·29-s + 2·30-s + 31-s − 32-s + 6·33-s + 8·34-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.447·5-s − 0.816·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.904·11-s + 0.577·12-s + 0.277·13-s − 0.516·15-s + 1/4·16-s − 1.94·17-s − 0.235·18-s − 0.223·20-s − 0.639·22-s − 0.408·24-s + 1/5·25-s − 0.196·26-s − 0.769·27-s − 0.742·29-s + 0.365·30-s + 0.179·31-s − 0.176·32-s + 1.04·33-s + 1.37·34-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 \( 1 - 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 5 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 13 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 - 9 T + p T^{2} \)
79 \( 1 - 3 T + p T^{2} \)
83 \( 1 + 5 T + p T^{2} \)
89 \( 1 - 7 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−8.286717042604170319017974851646, −7.54709382746671700185350371041, −6.79398657266478159723759092862, −6.22218292835765381830851614727, −4.93661172271683553385092184203, −3.95831677854594451407482295960, −3.36012003968007557369367751778, −2.34976718606610478933462474589, −1.58960418822414296362850646677, 0, 1.58960418822414296362850646677, 2.34976718606610478933462474589, 3.36012003968007557369367751778, 3.95831677854594451407482295960, 4.93661172271683553385092184203, 6.22218292835765381830851614727, 6.79398657266478159723759092862, 7.54709382746671700185350371041, 8.286717042604170319017974851646

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