Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 \cdot 17^{2} $
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 − 3-s − 4-s − 5-s − 6-s − 7-s − 3·8-s + 9-s − 10-s + 12-s − 6·13-s − 14-s + 15-s − 16-s + 18-s − 8·19-s + 20-s + 21-s − 8·23-s + 3·24-s + 25-s − 6·26-s − 27-s + 28-s + 2·29-s + 30-s − 4·31-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 − 1.06·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s − 1.66·13-s − 0.267·14-s + 0.258·15-s − 1/4·16-s + 0.235·18-s − 1.83·19-s + 0.223·20-s + 0.218·21-s − 1.66·23-s + 0.612·24-s + 1/5·25-s − 1.17·26-s − 0.192·27-s + 0.188·28-s + 0.371·29-s + 0.182·30-s − 0.718·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 30345 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30345 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(30345\)    =    \(3 \cdot 5 \cdot 7 \cdot 17^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{30345} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 30345,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7,\;17\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;7,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 + T \)
17 \( 1 \)
good2 \( 1 - T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−15.49598016239850, −14.60034294592895, −14.48539688120156, −13.87063109479439, −13.04627009146062, −12.69379685134684, −12.33695130673866, −11.89964924869463, −11.28014831837275, −10.51988342879034, −10.06005349160190, −9.569873676548730, −8.862950337541167, −8.348163525462949, −7.590386819186511, −7.089746417447654, −6.297281252383969, −5.904994525235037, −5.234279455575574, −4.601191680893936, −4.085893493629938, −3.730424258379182, −2.538962345519273, −2.216521207480749, −0.6551613151337828, 0, 0.6551613151337828, 2.216521207480749, 2.538962345519273, 3.730424258379182, 4.085893493629938, 4.601191680893936, 5.234279455575574, 5.904994525235037, 6.297281252383969, 7.089746417447654, 7.590386819186511, 8.348163525462949, 8.862950337541167, 9.569873676548730, 10.06005349160190, 10.51988342879034, 11.28014831837275, 11.89964924869463, 12.33695130673866, 12.69379685134684, 13.04627009146062, 13.87063109479439, 14.48539688120156, 14.60034294592895, 15.49598016239850

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