Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 \cdot 17^{2} $
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 + 3·8-s + 9-s + 10-s + 2·11-s − 12-s + 2·13-s + 14-s − 15-s − 16-s − 18-s − 2·19-s + 20-s − 21-s − 2·22-s − 2·23-s + 3·24-s + 25-s − 2·26-s + 27-s + 28-s − 10·29-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.377·7-s + 1.06·8-s + 1/3·9-s + 0.316·10-s + 0.603·11-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 − 0.458·19-s + 0.223·20-s − 0.218·21-s − 0.426·22-s − 0.417·23-s + 0.612·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s + 0.188·28-s − 1.85·29-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  =  0
Selberg data  =  $(2,\ 30345,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.6335338113$
$L(\frac12)$  $\approx$  $0.6335338113$
$L(\frac{3}{2})$   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 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
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

−14.97877405956478, −14.68151695490527, −14.13257191844197, −13.41105718834727, −13.11343484119576, −12.65101584594402, −11.84247390574492, −11.30283669699716, −10.72425412497052, −10.13193660965451, −9.597063737577515, −9.113770931980259, −8.612850634302678, −8.242149644894991, −7.524506563149172, −7.063088532734723, −6.443783945878125, −5.551111365777007, −4.982629465952741, −4.021727436807332, −3.782095996705465, −3.175519787005382, −1.891297966099057, −1.562825907569371, −0.3339125773110024, 0.3339125773110024, 1.562825907569371, 1.891297966099057, 3.175519787005382, 3.782095996705465, 4.021727436807332, 4.982629465952741, 5.551111365777007, 6.443783945878125, 7.063088532734723, 7.524506563149172, 8.242149644894991, 8.612850634302678, 9.113770931980259, 9.597063737577515, 10.13193660965451, 10.72425412497052, 11.30283669699716, 11.84247390574492, 12.65101584594402, 13.11343484119576, 13.41105718834727, 14.13257191844197, 14.68151695490527, 14.97877405956478

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