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-s − 3-s − 4-s + 5-s + 6-s + 7-s + 3·8-s + 9-s − 10-s − 2·11-s + 12-s + 2·13-s − 14-s − 15-s − 16-s − 18-s − 2·19-s − 20-s − 21-s + 2·22-s + 2·23-s − 3·24-s + 25-s − 2·26-s − 27-s − 28-s − 6·29-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 − 0.603·11-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 − 0.458·19-s − 0.223·20-s − 0.218·21-s + 0.426·22-s + 0.417·23-s − 0.612·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s − 0.188·28-s − 1.11·29-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  =  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 \( 1 + T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−15.57953723610804, −14.71399739897203, −14.45832254740599, −13.62709619450417, −13.26126451225818, −12.84422255713016, −12.24408288261400, −11.45501493853361, −10.89016435522162, −10.59060081108352, −10.07994999581347, −9.369713734593923, −8.928883718171462, −8.453617918695730, −7.747769892485505, −7.237735982808060, −6.670032470941877, −5.711255218640653, −5.397553138149155, −4.850741570911944, −4.009645674508384, −3.528521238733668, −2.281356177701838, −1.715335906161676, −0.8745184047106213, 0, 0.8745184047106213, 1.715335906161676, 2.281356177701838, 3.528521238733668, 4.009645674508384, 4.850741570911944, 5.397553138149155, 5.711255218640653, 6.670032470941877, 7.237735982808060, 7.747769892485505, 8.453617918695730, 8.928883718171462, 9.369713734593923, 10.07994999581347, 10.59060081108352, 10.89016435522162, 11.45501493853361, 12.24408288261400, 12.84422255713016, 13.26126451225818, 13.62709619450417, 14.45832254740599, 14.71399739897203, 15.57953723610804

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