Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $0.103 - 0.994i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 + 1.22i)2-s + (1.5 + 0.866i)3-s + (−0.999 + 1.73i)4-s + (1.93 − 1.11i)5-s + 2.44i·6-s + (5.10 + 4.79i)7-s − 2.82·8-s + (1.5 + 2.59i)9-s + (2.73 + 1.58i)10-s + (0.919 − 1.59i)11-s + (−2.99 + 1.73i)12-s + 5.40i·13-s + (−2.26 + 9.63i)14-s + 3.87·15-s + (−2.00 − 3.46i)16-s + (8.71 + 5.02i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.5 + 0.288i)3-s + (−0.249 + 0.433i)4-s + (0.387 − 0.223i)5-s + 0.408i·6-s + (0.728 + 0.684i)7-s − 0.353·8-s + (0.166 + 0.288i)9-s + (0.273 + 0.158i)10-s + (0.0835 − 0.144i)11-s + (−0.249 + 0.144i)12-s + 0.415i·13-s + (−0.161 + 0.688i)14-s + 0.258·15-s + (−0.125 − 0.216i)16-s + (0.512 + 0.295i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.103 - 0.994i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.103 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.103 - 0.994i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (31, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ 0.103 - 0.994i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.72298 + 1.55294i\)
\(L(\frac12)\)  \(\approx\)  \(1.72298 + 1.55294i\)
\(L(2)\)   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 \( 1 + (-0.707 - 1.22i)T \)
3 \( 1 + (-1.5 - 0.866i)T \)
5 \( 1 + (-1.93 + 1.11i)T \)
7 \( 1 + (-5.10 - 4.79i)T \)
good11 \( 1 + (-0.919 + 1.59i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 5.40iT - 169T^{2} \)
17 \( 1 + (-8.71 - 5.02i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (7.96 - 4.59i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-0.460 - 0.797i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 - 12.5T + 841T^{2} \)
31 \( 1 + (36.1 + 20.8i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (3.64 + 6.30i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 52.3iT - 1.68e3T^{2} \)
43 \( 1 + 8.12T + 1.84e3T^{2} \)
47 \( 1 + (-29.4 + 16.9i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-52.0 + 90.1i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-12.5 - 7.26i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-20.5 + 11.8i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-7.46 + 12.9i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 17.9T + 5.04e3T^{2} \)
73 \( 1 + (107. + 62.1i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-40.4 - 70.0i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 154. iT - 6.88e3T^{2} \)
89 \( 1 + (-58.9 + 34.0i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 88.1iT - 9.40e3T^{2} \)
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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

−12.52030718579872585267073101364, −11.55847843963443843462378811560, −10.29740872466506484862339480236, −9.089396780639354044490515236916, −8.466613061485183875347145564546, −7.37740350400127627603624829971, −5.97707465412467643458967092126, −5.05206445565510467516714231971, −3.79882969927901047404357595070, −2.10498006632038874405151502940, 1.32873314219706409686147245452, 2.78725387599438033204172849071, 4.14854185842066092438285550327, 5.40679356147192571372529421036, 6.84143380173565019352715442682, 7.905606467825844605272081804082, 9.053651492908161167561698530032, 10.16286227359566795383774764080, 10.90278689580544019622415408684, 11.97763924417380998063981208352

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