Properties

Degree $2$
Conductor $210$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s + 5-s − 6-s + 7-s − 8-s + 9-s − 10-s + 12-s + 2·13-s − 14-s + 15-s + 16-s − 6·17-s − 18-s + 8·19-s + 20-s + 21-s − 24-s + 25-s − 2·26-s + 27-s + 28-s + 6·29-s − 30-s − 4·31-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 − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s + 0.554·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 1.83·19-s + 0.223·20-s + 0.218·21-s − 0.204·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s + 0.188·28-s + 1.11·29-s − 0.182·30-s − 0.718·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{210} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17666\)
\(L(\frac12)\) \(\approx\) \(1.17666\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 - T \)
good11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−12.25378438340330619534834471577, −11.21669057172523093750433544809, −10.29726946096077498624836812705, −9.262397060563074490443203035163, −8.581528352643131576230344729464, −7.48230039465939347417283743898, −6.43968643454495280602009146366, −4.97103303999261607614028493172, −3.22977772739188491587716942834, −1.69594451483764738863265937021, 1.69594451483764738863265937021, 3.22977772739188491587716942834, 4.97103303999261607614028493172, 6.43968643454495280602009146366, 7.48230039465939347417283743898, 8.581528352643131576230344729464, 9.262397060563074490443203035163, 10.29726946096077498624836812705, 11.21669057172523093750433544809, 12.25378438340330619534834471577

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