Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 \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 + 7-s + 8-s + 9-s − 10-s + 12-s + 2·13-s + 14-s − 15-s + 16-s − 17-s + 18-s − 4·19-s − 20-s + 21-s + 24-s + 25-s + 2·26-s + 27-s + 28-s + 6·29-s − 30-s + 8·31-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s + 0.554·13-s + 0.267·14-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 + 0.218·21-s + 0.204·24-s + 1/5·25-s + 0.392·26-s + 0.192·27-s + 0.188·28-s + 1.11·29-s − 0.182·30-s + 1.43·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 3570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 3570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3570\)    =    \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 17\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{3570} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 3570,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.794908686$
$L(\frac12)$  $\approx$  $3.794908686$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;7,\;17\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;17\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
3 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 - T \)
17 \( 1 + T \)
good11 \( 1 + p T^{2} \)
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 \( 1 - 2 T + p T^{2} \)
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 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−18.41697474983958, −17.65618958309607, −17.06836603028749, −16.14352026419367, −15.75649379989061, −15.09954346052697, −14.58604918783801, −13.95036239238552, −13.33148738356287, −12.79982841149120, −11.92605818756681, −11.56345428633944, −10.56533229026447, −10.24037661581654, −9.027044037310383, −8.479474614302360, −7.863472021123861, −7.024668699941976, −6.350749304359889, −5.498587872124513, −4.373689081198981, −4.166662028602177, −3.011388340146359, −2.308451498492852, −1.055612249682634, 1.055612249682634, 2.308451498492852, 3.011388340146359, 4.166662028602177, 4.373689081198981, 5.498587872124513, 6.350749304359889, 7.024668699941976, 7.863472021123861, 8.479474614302360, 9.027044037310383, 10.24037661581654, 10.56533229026447, 11.56345428633944, 11.92605818756681, 12.79982841149120, 13.33148738356287, 13.95036239238552, 14.58604918783801, 15.09954346052697, 15.75649379989061, 16.14352026419367, 17.06836603028749, 17.65618958309607, 18.41697474983958

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