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 − 4·11-s − 12-s + 2·13-s + 14-s − 15-s − 16-s − 18-s + 8·19-s + 20-s − 21-s + 4·22-s + 4·23-s + 3·24-s + 25-s − 2·26-s + 27-s + 28-s − 2·29-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 − 1.20·11-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 + 1.83·19-s + 0.223·20-s − 0.218·21-s + 0.852·22-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 − 0.371·29-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  =  $(2,\ 30345,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.623897044$
$L(\frac12)$  $\approx$  $1.623897044$
$L(\frac{3}{2})$   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 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
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

−15.24941929621141, −14.59174302499039, −13.88584226809694, −13.58853789642048, −13.13686389137369, −12.59060809486000, −11.92706149149872, −11.23694708309186, −10.68046259674638, −10.13726270361092, −9.690884078053824, −9.098545251697402, −8.652323319520855, −8.053182726472878, −7.543541455556844, −7.257687345600027, −6.364178302794175, −5.394327916112682, −5.095410789474352, −4.235995688992315, −3.631060597103828, −2.967972962550732, −2.321028637381559, −1.111804927507917, −0.6503096275039656, 0.6503096275039656, 1.111804927507917, 2.321028637381559, 2.967972962550732, 3.631060597103828, 4.235995688992315, 5.095410789474352, 5.394327916112682, 6.364178302794175, 7.257687345600027, 7.543541455556844, 8.053182726472878, 8.652323319520855, 9.098545251697402, 9.690884078053824, 10.13726270361092, 10.68046259674638, 11.23694708309186, 11.92706149149872, 12.59060809486000, 13.13686389137369, 13.58853789642048, 13.88584226809694, 14.59174302499039, 15.24941929621141

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