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 − 2·13-s + 14-s + 15-s − 16-s − 18-s − 20-s − 21-s − 4·23-s + 3·24-s + 25-s + 2·26-s + 27-s + 28-s + 6·29-s − 30-s − 5·32-s − 35-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 − 0.554·13-s + 0.267·14-s + 0.258·15-s − 1/4·16-s − 0.235·18-s − 0.223·20-s − 0.218·21-s − 0.834·23-s + 0.612·24-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 − 0.883·32-s − 0.169·35-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 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
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 \( 1 - 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
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

−15.38179299224084, −14.82559069533775, −14.24202506390808, −13.72176793219196, −13.46658162121180, −12.86379601592065, −12.14735473095220, −11.84095779216562, −10.68834135295585, −10.45674896242216, −9.808922731134307, −9.528789591130948, −8.892311709783544, −8.411084175861303, −7.885736816833070, −7.322622569536564, −6.634943713226651, −6.073668740103516, −5.120917119224832, −4.759585242895457, −3.942000193392801, −3.330216591184412, −2.474118946932945, −1.828874148346721, −0.9734027505019840, 0, 0.9734027505019840, 1.828874148346721, 2.474118946932945, 3.330216591184412, 3.942000193392801, 4.759585242895457, 5.120917119224832, 6.073668740103516, 6.634943713226651, 7.322622569536564, 7.885736816833070, 8.411084175861303, 8.892311709783544, 9.528789591130948, 9.808922731134307, 10.45674896242216, 10.68834135295585, 11.84095779216562, 12.14735473095220, 12.86379601592065, 13.46658162121180, 13.72176793219196, 14.24202506390808, 14.82559069533775, 15.38179299224084

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