Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 11 \cdot 17 $
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-s + 3-s + 4-s + 5-s − 6-s − 4·7-s − 8-s + 9-s − 10-s + 11-s + 12-s + 2·13-s + 4·14-s + 15-s + 16-s − 17-s − 18-s − 4·19-s + 20-s − 4·21-s − 22-s − 24-s + 25-s − 2·26-s + 27-s − 4·28-s − 6·29-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s + 0.288·12-s + 0.554·13-s + 1.06·14-s + 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.917·19-s + 0.223·20-s − 0.872·21-s − 0.213·22-s − 0.204·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s − 0.755·28-s − 1.11·29-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(5610\)    =    \(2 \cdot 3 \cdot 5 \cdot 11 \cdot 17\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{5610} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 5610,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.443030053$
$L(\frac12)$  $\approx$  $1.443030053$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;11,\;17\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;11,\;17\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
3 \( 1 - T \)
5 \( 1 - T \)
11 \( 1 - T \)
17 \( 1 + T \)
good7 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−17.48030821827289, −16.95902716303428, −16.43544172405010, −15.64481669275733, −15.47350761933494, −14.58090988597649, −13.86506540270328, −13.28030375100515, −12.79020356606408, −12.14442130543614, −11.31019917602474, −10.44382602693986, −10.08989627210992, −9.428907338553515, −8.801689045031579, −8.466098266741054, −7.345799858354735, −6.801557049059003, −6.218972167541508, −5.539116594957834, −4.194032010050571, −3.529307671431906, −2.688036720757664, −1.931423502675833, −0.6757425900965431, 0.6757425900965431, 1.931423502675833, 2.688036720757664, 3.529307671431906, 4.194032010050571, 5.539116594957834, 6.218972167541508, 6.801557049059003, 7.345799858354735, 8.466098266741054, 8.801689045031579, 9.428907338553515, 10.08989627210992, 10.44382602693986, 11.31019917602474, 12.14442130543614, 12.79020356606408, 13.28030375100515, 13.86506540270328, 14.58090988597649, 15.47350761933494, 15.64481669275733, 16.43544172405010, 16.95902716303428, 17.48030821827289

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