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 − 11-s + 12-s − 13-s − 14-s + 15-s − 16-s + 18-s + 20-s + 21-s − 22-s + 3·24-s + 25-s − 26-s − 27-s + 28-s + 6·29-s + 30-s − 7·31-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 − 0.301·11-s + 0.288·12-s − 0.277·13-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 − 0.213·22-s + 0.612·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s + 0.188·28-s + 1.11·29-s + 0.182·30-s − 1.25·31-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  =  0
Selberg data  =  $(2,\ 30345,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.5256880456$
$L(\frac12)$  $\approx$  $0.5256880456$
$L(\frac{3}{2})$   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 \( 1 + T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 - 13 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 5 T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
97 \( 1 + 13 T + p T^{2} \)
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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

−15.04760097065735, −14.51694499337504, −14.11219573029344, −13.40635136120420, −13.00717849187956, −12.44167606366997, −12.14819711921305, −11.50962376756458, −11.00044260457794, −10.13407245958016, −10.00042758132407, −9.071320308562683, −8.676924325851866, −8.023907114473164, −7.262033744843531, −6.728482846748594, −6.156055564298773, −5.298810606883698, −5.180568770395625, −4.363666508334248, −3.793753439381438, −3.221253451358022, −2.469805984927891, −1.359972964623774, −0.2667202294533952, 0.2667202294533952, 1.359972964623774, 2.469805984927891, 3.221253451358022, 3.793753439381438, 4.363666508334248, 5.180568770395625, 5.298810606883698, 6.156055564298773, 6.728482846748594, 7.262033744843531, 8.023907114473164, 8.676924325851866, 9.071320308562683, 10.00042758132407, 10.13407245958016, 11.00044260457794, 11.50962376756458, 12.14819711921305, 12.44167606366997, 13.00717849187956, 13.40635136120420, 14.11219573029344, 14.51694499337504, 15.04760097065735

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