Properties

Degree $2$
Conductor $304920$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 7-s + 2·13-s + 2·17-s + 4·19-s + 25-s − 10·29-s − 35-s + 6·37-s − 6·41-s + 4·43-s + 8·47-s + 49-s − 6·53-s + 4·59-s + 10·61-s − 2·65-s + 4·67-s + 16·71-s + 14·73-s − 8·79-s − 4·83-s − 2·85-s − 10·89-s + 2·91-s − 4·95-s + 10·97-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.377·7-s + 0.554·13-s + 0.485·17-s + 0.917·19-s + 1/5·25-s − 1.85·29-s − 0.169·35-s + 0.986·37-s − 0.937·41-s + 0.609·43-s + 1.16·47-s + 1/7·49-s − 0.824·53-s + 0.520·59-s + 1.28·61-s − 0.248·65-s + 0.488·67-s + 1.89·71-s + 1.63·73-s − 0.900·79-s − 0.439·83-s − 0.216·85-s − 1.05·89-s + 0.209·91-s − 0.410·95-s + 1.01·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304920\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{304920} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 304920,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + T \)
7 \( 1 - T \)
11 \( 1 \)
good13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.91626304084082, −12.45066938756519, −11.91389693361757, −11.49024841754307, −11.03930230653752, −10.90277651398510, −10.01085847213404, −9.771219133805598, −9.201160099589314, −8.778927577715295, −8.166233600708190, −7.820415503257223, −7.420720534539275, −6.901695913136545, −6.356825831237832, −5.743041347597301, −5.282579639987317, −4.997971307592247, −4.031710128132975, −3.881763950046987, −3.345154667150756, −2.611585444558892, −2.099814397494389, −1.296101037314817, −0.8810046814437449, 0, 0.8810046814437449, 1.296101037314817, 2.099814397494389, 2.611585444558892, 3.345154667150756, 3.881763950046987, 4.031710128132975, 4.997971307592247, 5.282579639987317, 5.743041347597301, 6.356825831237832, 6.901695913136545, 7.420720534539275, 7.820415503257223, 8.166233600708190, 8.778927577715295, 9.201160099589314, 9.771219133805598, 10.01085847213404, 10.90277651398510, 11.03930230653752, 11.49024841754307, 11.91389693361757, 12.45066938756519, 12.91626304084082

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