Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $0.975 - 0.220i$
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 + (−2.45 + 1.72i)3-s + (−1.73 + i)4-s + (2.65 − 4.23i)5-s + (−3.25 − 2.72i)6-s + (3.47 − 6.07i)7-s + (−2 − 1.99i)8-s + (3.04 − 8.46i)9-s + (6.75 + 2.07i)10-s + (5.23 − 3.02i)11-s + (2.52 − 5.44i)12-s + (7.33 + 7.33i)13-s + (9.57 + 2.51i)14-s + (0.787 + 14.9i)15-s + (1.99 − 3.46i)16-s + (6.03 − 22.5i)17-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (−0.818 + 0.575i)3-s + (−0.433 + 0.250i)4-s + (0.531 − 0.847i)5-s + (−0.542 − 0.453i)6-s + (0.495 − 0.868i)7-s + (−0.250 − 0.249i)8-s + (0.338 − 0.940i)9-s + (0.675 + 0.207i)10-s + (0.475 − 0.274i)11-s + (0.210 − 0.453i)12-s + (0.564 + 0.564i)13-s + (0.683 + 0.179i)14-s + (0.0524 + 0.998i)15-s + (0.124 − 0.216i)16-s + (0.354 − 1.32i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.220i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.975 - 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.975 - 0.220i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (47, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ 0.975 - 0.220i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.41847 + 0.158399i\)
\(L(\frac12)\)  \(\approx\)  \(1.41847 + 0.158399i\)
\(L(2)\)   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 \( 1 + (2.45 - 1.72i)T \)
5 \( 1 + (-2.65 + 4.23i)T \)
7 \( 1 + (-3.47 + 6.07i)T \)
good11 \( 1 + (-5.23 + 3.02i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-7.33 - 7.33i)T + 169iT^{2} \)
17 \( 1 + (-6.03 + 22.5i)T + (-250. - 144.5i)T^{2} \)
19 \( 1 + (17.0 - 29.6i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-26.5 + 7.11i)T + (458. - 264.5i)T^{2} \)
29 \( 1 + 6.86T + 841T^{2} \)
31 \( 1 + (-38.9 + 22.4i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-34.1 + 9.14i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 + 18.2T + 1.68e3T^{2} \)
43 \( 1 + (-17.2 + 17.2i)T - 1.84e3iT^{2} \)
47 \( 1 + (23.3 - 6.24i)T + (1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (-20.7 + 77.2i)T + (-2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (97.9 - 56.5i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-16.8 - 9.74i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (2.92 - 10.9i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 - 80.6iT - 5.04e3T^{2} \)
73 \( 1 + (0.519 - 1.93i)T + (-4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (-73.1 - 42.2i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-4.56 + 4.56i)T - 6.88e3iT^{2} \)
89 \( 1 + (21.5 + 12.4i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (38.4 - 38.4i)T - 9.40e3iT^{2} \)
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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

−12.16033342462045084122988046228, −11.26416604830300289623920631765, −10.11471952569450483529252388134, −9.253307638843028863015272035853, −8.187635405429352608862910496494, −6.79755544622769649884845678941, −5.85434736371424219473060325500, −4.78700337208283945195784167248, −3.97451781755268286931556522093, −0.986465525332442916081941819871, 1.52573815029914892009946866006, 2.83268201815007338197663145794, 4.72655599379931831251934725065, 5.89273114090757063592517955352, 6.66036867723748683190644349204, 8.134839069335552729375495255788, 9.332303168759996480227909351932, 10.69997912382427548373610388314, 11.01193617553932635143160744206, 12.06002904068032012567205994402

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