Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $-1$
Motivic weight 3
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3·3-s + 4·4-s − 5·5-s + 6·6-s + 7·7-s − 8·8-s + 9·9-s + 10·10-s + 12·11-s − 12·12-s + 2·13-s − 14·14-s + 15·15-s + 16·16-s − 18·17-s − 18·18-s + 56·19-s − 20·20-s − 21·21-s − 24·22-s − 156·23-s + 24·24-s + 25·25-s − 4·26-s − 27·27-s + 28·28-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 + 0.328·11-s − 0.288·12-s + 0.0426·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.256·17-s − 0.235·18-s + 0.676·19-s − 0.223·20-s − 0.218·21-s − 0.232·22-s − 1.41·23-s + 0.204·24-s + 1/5·25-s − 0.0301·26-s − 0.192·27-s + 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(3\)
character  :  $\chi_{210} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 210,\ (\ :3/2),\ -1)\)
\(L(2)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{5}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + p T \)
3 \( 1 + p T \)
5 \( 1 + p T \)
7 \( 1 - p T \)
good11 \( 1 - 12 T + p^{3} T^{2} \)
13 \( 1 - 2 T + p^{3} T^{2} \)
17 \( 1 + 18 T + p^{3} T^{2} \)
19 \( 1 - 56 T + p^{3} T^{2} \)
23 \( 1 + 156 T + p^{3} T^{2} \)
29 \( 1 + 186 T + p^{3} T^{2} \)
31 \( 1 + 52 T + p^{3} T^{2} \)
37 \( 1 + 178 T + p^{3} T^{2} \)
41 \( 1 + 138 T + p^{3} T^{2} \)
43 \( 1 + 412 T + p^{3} T^{2} \)
47 \( 1 + 456 T + p^{3} T^{2} \)
53 \( 1 + 198 T + p^{3} T^{2} \)
59 \( 1 - 348 T + p^{3} T^{2} \)
61 \( 1 - 110 T + p^{3} T^{2} \)
67 \( 1 + 196 T + p^{3} T^{2} \)
71 \( 1 + 936 T + p^{3} T^{2} \)
73 \( 1 - 542 T + p^{3} T^{2} \)
79 \( 1 - 992 T + p^{3} T^{2} \)
83 \( 1 + 276 T + p^{3} T^{2} \)
89 \( 1 - 630 T + p^{3} T^{2} \)
97 \( 1 - 110 T + p^{3} T^{2} \)
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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

−11.50349896476997479582438801378, −10.47295122931231392990389360056, −9.531357922044252908352881557464, −8.376305907423197518454620966567, −7.45702679743899948770516904639, −6.39580326601707841263006404860, −5.13927132311198187264103254368, −3.67073875116458922916875141602, −1.69965255347246301006725902367, 0, 1.69965255347246301006725902367, 3.67073875116458922916875141602, 5.13927132311198187264103254368, 6.39580326601707841263006404860, 7.45702679743899948770516904639, 8.376305907423197518454620966567, 9.531357922044252908352881557464, 10.47295122931231392990389360056, 11.50349896476997479582438801378

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