Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $0.967 - 0.252i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.366 − 1.36i)2-s + (−0.448 + 1.67i)3-s + (−1.73 + i)4-s + (−3.76 − 3.28i)5-s + 2.44·6-s + (5.65 + 4.13i)7-s + (2 + 1.99i)8-s + (−2.59 − 1.50i)9-s + (−3.11 + 6.34i)10-s + (2.47 + 4.28i)11-s + (−0.896 − 3.34i)12-s + (7.82 + 7.82i)13-s + (3.57 − 9.23i)14-s + (7.18 − 4.82i)15-s + (1.99 − 3.46i)16-s + (3.41 + 0.914i)17-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (−0.149 + 0.557i)3-s + (−0.433 + 0.250i)4-s + (−0.753 − 0.657i)5-s + 0.408·6-s + (0.807 + 0.590i)7-s + (0.250 + 0.249i)8-s + (−0.288 − 0.166i)9-s + (−0.311 + 0.634i)10-s + (0.224 + 0.389i)11-s + (−0.0747 − 0.278i)12-s + (0.602 + 0.602i)13-s + (0.255 − 0.659i)14-s + (0.479 − 0.321i)15-s + (0.124 − 0.216i)16-s + (0.200 + 0.0537i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.252i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.967 - 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.967 - 0.252i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (163, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ 0.967 - 0.252i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.19798 + 0.153715i\)
\(L(\frac12)\)  \(\approx\)  \(1.19798 + 0.153715i\)
\(L(2)\)   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 \( 1 + (0.366 + 1.36i)T \)
3 \( 1 + (0.448 - 1.67i)T \)
5 \( 1 + (3.76 + 3.28i)T \)
7 \( 1 + (-5.65 - 4.13i)T \)
good11 \( 1 + (-2.47 - 4.28i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (-7.82 - 7.82i)T + 169iT^{2} \)
17 \( 1 + (-3.41 - 0.914i)T + (250. + 144.5i)T^{2} \)
19 \( 1 + (-26.8 - 15.5i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-17.2 + 4.61i)T + (458. - 264.5i)T^{2} \)
29 \( 1 - 24.0iT - 841T^{2} \)
31 \( 1 + (-7.79 - 13.5i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (8.92 + 33.2i)T + (-1.18e3 + 684.5i)T^{2} \)
41 \( 1 + 19.3T + 1.68e3T^{2} \)
43 \( 1 + (11.5 + 11.5i)T + 1.84e3iT^{2} \)
47 \( 1 + (-8.13 - 30.3i)T + (-1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (-26.9 + 100. i)T + (-2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (-13.5 + 7.81i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (44.1 - 76.5i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (75.6 + 20.2i)T + (3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + 59.7T + 5.04e3T^{2} \)
73 \( 1 + (27.5 - 102. i)T + (-4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (23.6 + 13.6i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (48.9 + 48.9i)T + 6.88e3iT^{2} \)
89 \( 1 + (-98.6 - 56.9i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-99.3 + 99.3i)T - 9.40e3iT^{2} \)
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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

−11.84875548438890773398707411956, −11.46488066602537546792423564463, −10.32885544703130511279990017078, −9.131433153796882441892488853615, −8.562565706571027654134487562188, −7.38874641365877936520822071564, −5.47902363504357934270836958647, −4.56183260947854419609383393243, −3.41612268765150110831424527864, −1.41274170846636405237494959365, 0.892446552804224083357942211937, 3.27157237700295744658238400346, 4.76897824837011310707623893883, 6.08246941777137154183514809404, 7.25330153942273992721265916489, 7.76909355543688287449908395869, 8.776666597833055002533235681077, 10.27067647296091239317742122438, 11.23547083695790020675799916808, 11.87241311342331293997264495025

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