Properties

Degree 2
Conductor 4013
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 2·3-s + 2·4-s + 4·5-s − 4·6-s + 7-s + 9-s + 8·10-s + 3·11-s − 4·12-s + 2·13-s + 2·14-s − 8·15-s − 4·16-s − 2·17-s + 2·18-s + 19-s + 8·20-s − 2·21-s + 6·22-s + 11·25-s + 4·26-s + 4·27-s + 2·28-s + 8·29-s − 16·30-s − 5·31-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.15·3-s + 4-s + 1.78·5-s − 1.63·6-s + 0.377·7-s + 1/3·9-s + 2.52·10-s + 0.904·11-s − 1.15·12-s + 0.554·13-s + 0.534·14-s − 2.06·15-s − 16-s − 0.485·17-s + 0.471·18-s + 0.229·19-s + 1.78·20-s − 0.436·21-s + 1.27·22-s + 11/5·25-s + 0.784·26-s + 0.769·27-s + 0.377·28-s + 1.48·29-s − 2.92·30-s − 0.898·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4013\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4013} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4013,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.224266485$
$L(\frac12)$  $\approx$  $4.224266485$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 4013$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p = 4013$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad4013 \( 1+O(T) \)
good2 \( 1 - p T + p T^{2} \)
3 \( 1 + 2 T + p T^{2} \)
5 \( 1 - 4 T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 - 7 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 13 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 17 T + p T^{2} \)
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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

−8.733307016292182201470530080278, −7.13195109010699529855883116032, −6.44847889143151588305592272408, −6.00987769206640373341160680249, −5.55374327593100525288801835894, −4.86892351817549506409155287668, −4.19801965133586384564074386093, −3.02865667935775685711300181536, −2.10489078586203324162174094622, −1.08014622018469319166471885891, 1.08014622018469319166471885891, 2.10489078586203324162174094622, 3.02865667935775685711300181536, 4.19801965133586384564074386093, 4.86892351817549506409155287668, 5.55374327593100525288801835894, 6.00987769206640373341160680249, 6.44847889143151588305592272408, 7.13195109010699529855883116032, 8.733307016292182201470530080278

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