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  + 3-s − 2·4-s − 5-s + 7-s + 9-s + 2·11-s − 2·12-s + 3·13-s − 15-s + 4·16-s − 2·19-s + 2·20-s + 21-s + 3·23-s + 25-s + 27-s − 2·28-s − 4·29-s + 5·31-s + 2·33-s − 35-s − 2·36-s + 37-s + 3·39-s − 11·41-s + 4·43-s − 4·44-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.603·11-s − 0.577·12-s + 0.832·13-s − 0.258·15-s + 16-s − 0.458·19-s + 0.447·20-s + 0.218·21-s + 0.625·23-s + 1/5·25-s + 0.192·27-s − 0.377·28-s − 0.742·29-s + 0.898·31-s + 0.348·33-s − 0.169·35-s − 1/3·36-s + 0.164·37-s + 0.480·39-s − 1.71·41-s + 0.609·43-s − 0.603·44-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 \( 1 + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 + 11 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 5 T + p T^{2} \)
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

−15.19011653043879, −14.89901447084892, −14.27754799433181, −13.80790762539706, −13.36828617634322, −12.86391578000931, −12.29361710363005, −11.68540690155468, −11.11089630345116, −10.53750840178595, −9.881528686668992, −9.338389010295830, −8.764916697590490, −8.434266888000671, −7.919040174965436, −7.285698416867676, −6.527259720658172, −5.969537013735330, −5.049742950458882, −4.647862448653398, −3.925477031738557, −3.528231276741998, −2.810527333568341, −1.678365850370996, −1.105628549436019, 0, 1.105628549436019, 1.678365850370996, 2.810527333568341, 3.528231276741998, 3.925477031738557, 4.647862448653398, 5.049742950458882, 5.969537013735330, 6.527259720658172, 7.285698416867676, 7.919040174965436, 8.434266888000671, 8.764916697590490, 9.338389010295830, 9.881528686668992, 10.53750840178595, 11.11089630345116, 11.68540690155468, 12.29361710363005, 12.86391578000931, 13.36828617634322, 13.80790762539706, 14.27754799433181, 14.89901447084892, 15.19011653043879

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