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  − 2·2-s + 3-s + 2·4-s − 5-s − 2·6-s + 7-s + 9-s + 2·10-s + 3·11-s + 2·12-s − 6·13-s − 2·14-s − 15-s − 4·16-s − 2·18-s − 19-s − 2·20-s + 21-s − 6·22-s + 25-s + 12·26-s + 27-s + 2·28-s − 5·29-s + 2·30-s − 8·31-s + 8·32-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.577·3-s + 4-s − 0.447·5-s − 0.816·6-s + 0.377·7-s + 1/3·9-s + 0.632·10-s + 0.904·11-s + 0.577·12-s − 1.66·13-s − 0.534·14-s − 0.258·15-s − 16-s − 0.471·18-s − 0.229·19-s − 0.447·20-s + 0.218·21-s − 1.27·22-s + 1/5·25-s + 2.35·26-s + 0.192·27-s + 0.377·28-s − 0.928·29-s + 0.365·30-s − 1.43·31-s + 1.41·32-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 \( 1 + p T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 + 11 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 - 15 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−15.39815908733262, −14.92334688762954, −14.29661988403789, −14.19986695967603, −13.15270342136692, −12.67764787669202, −12.08663694386909, −11.47528535443602, −11.00196840064523, −10.45957801423225, −9.750713552145864, −9.302711620655586, −9.084185270681167, −8.297393940301344, −7.838301421815433, −7.226968467923554, −7.115405668586070, −6.157005933389037, −5.264032807722800, −4.525115315483028, −4.044003857501402, −3.188885951242071, −2.249484499354134, −1.853975503732718, −0.8987737127439721, 0, 0.8987737127439721, 1.853975503732718, 2.249484499354134, 3.188885951242071, 4.044003857501402, 4.525115315483028, 5.264032807722800, 6.157005933389037, 7.115405668586070, 7.226968467923554, 7.838301421815433, 8.297393940301344, 9.084185270681167, 9.302711620655586, 9.750713552145864, 10.45957801423225, 11.00196840064523, 11.47528535443602, 12.08663694386909, 12.67764787669202, 13.15270342136692, 14.19986695967603, 14.29661988403789, 14.92334688762954, 15.39815908733262

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