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 + 12-s − 6·13-s + 14-s − 15-s − 16-s − 18-s + 4·19-s − 20-s + 21-s − 3·24-s + 25-s + 6·26-s − 27-s + 28-s + 2·29-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 − 1.66·13-s + 0.267·14-s − 0.258·15-s − 1/4·16-s − 0.235·18-s + 0.917·19-s − 0.223·20-s + 0.218·21-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 − 0.883·32-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  =  0
Selberg data  =  $(2,\ 30345,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.9673832101$
$L(\frac12)$  $\approx$  $0.9673832101$
$L(\frac{3}{2})$   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 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
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

−14.93855652852538, −14.73727468268986, −13.93398833355242, −13.61038722709455, −12.89162175086413, −12.50854956518765, −11.94641169984595, −11.32976900667891, −10.61928299826993, −10.15619624147581, −9.730826526452549, −9.277194622159613, −8.854023904129364, −7.805562289821918, −7.604716700896733, −6.962943698644965, −6.272476128517237, −5.481670416739031, −5.135130452805737, −4.428573588818876, −3.852885328914742, −2.750774481678863, −2.215351214118109, −1.096616117920754, −0.5270272641305597, 0.5270272641305597, 1.096616117920754, 2.215351214118109, 2.750774481678863, 3.852885328914742, 4.428573588818876, 5.135130452805737, 5.481670416739031, 6.272476128517237, 6.962943698644965, 7.604716700896733, 7.805562289821918, 8.854023904129364, 9.277194622159613, 9.730826526452549, 10.15619624147581, 10.61928299826993, 11.32976900667891, 11.94641169984595, 12.50854956518765, 12.89162175086413, 13.61038722709455, 13.93398833355242, 14.73727468268986, 14.93855652852538

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