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 + 10·29-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 + 1.85·29-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 \( 1 - 10 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
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

−15.39769399503127, −14.97650084157026, −14.21569777388004, −13.73685356872390, −13.20483017725646, −12.93952885559845, −12.08566552719830, −11.58491884693598, −10.95949323993726, −10.43869202842877, −9.990053797931395, −9.659973006249086, −8.709904679658086, −8.386906639459701, −7.984262341735118, −7.156450772929563, −6.508074667512208, −6.040361818839471, −5.167866077460691, −4.747701137405186, −4.308048049384909, −3.268733367454255, −2.504015775439587, −1.500429102504922, −0.9926449006458294, 0, 0.9926449006458294, 1.500429102504922, 2.504015775439587, 3.268733367454255, 4.308048049384909, 4.747701137405186, 5.167866077460691, 6.040361818839471, 6.508074667512208, 7.156450772929563, 7.984262341735118, 8.386906639459701, 8.709904679658086, 9.659973006249086, 9.990053797931395, 10.43869202842877, 10.95949323993726, 11.58491884693598, 12.08566552719830, 12.93952885559845, 13.20483017725646, 13.73685356872390, 14.21569777388004, 14.97650084157026, 15.39769399503127

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