Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $-0.772 - 0.634i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.36 + 0.366i)2-s + (0.145 + 2.99i)3-s + (1.73 + i)4-s + (−4.92 + 0.860i)5-s + (−0.898 + 4.14i)6-s + (1.68 + 6.79i)7-s + (1.99 + 2i)8-s + (−8.95 + 0.871i)9-s + (−7.04 − 0.627i)10-s + (−13.7 − 7.94i)11-s + (−2.74 + 5.33i)12-s + (5.32 + 5.32i)13-s + (−0.185 + 9.89i)14-s + (−3.29 − 14.6i)15-s + (1.99 + 3.46i)16-s + (16.9 − 4.55i)17-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (0.0484 + 0.998i)3-s + (0.433 + 0.250i)4-s + (−0.985 + 0.172i)5-s + (−0.149 + 0.691i)6-s + (0.240 + 0.970i)7-s + (0.249 + 0.250i)8-s + (−0.995 + 0.0968i)9-s + (−0.704 − 0.0627i)10-s + (−1.25 − 0.722i)11-s + (−0.228 + 0.444i)12-s + (0.409 + 0.409i)13-s + (−0.0132 + 0.706i)14-s + (−0.219 − 0.975i)15-s + (0.124 + 0.216i)16-s + (0.999 − 0.267i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.772 - 0.634i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.772 - 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.772 - 0.634i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (17, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ -0.772 - 0.634i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.561882 + 1.56903i\)
\(L(\frac12)\)  \(\approx\)  \(0.561882 + 1.56903i\)
\(L(2)\)   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 \( 1 + (-1.36 - 0.366i)T \)
3 \( 1 + (-0.145 - 2.99i)T \)
5 \( 1 + (4.92 - 0.860i)T \)
7 \( 1 + (-1.68 - 6.79i)T \)
good11 \( 1 + (13.7 + 7.94i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-5.32 - 5.32i)T + 169iT^{2} \)
17 \( 1 + (-16.9 + 4.55i)T + (250. - 144.5i)T^{2} \)
19 \( 1 + (-5.62 - 9.74i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (10.4 - 39.1i)T + (-458. - 264.5i)T^{2} \)
29 \( 1 + 22.8T + 841T^{2} \)
31 \( 1 + (-36.0 - 20.8i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-7.72 + 28.8i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 - 55.0T + 1.68e3T^{2} \)
43 \( 1 + (14.2 - 14.2i)T - 1.84e3iT^{2} \)
47 \( 1 + (6.77 - 25.2i)T + (-1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (-59.0 + 15.8i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (-10.7 - 6.18i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-14.5 + 8.40i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-103. + 27.8i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + 34.7iT - 5.04e3T^{2} \)
73 \( 1 + (56.8 - 15.2i)T + (4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (-40.5 + 23.4i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-75.6 + 75.6i)T - 6.88e3iT^{2} \)
89 \( 1 + (116. - 67.1i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (75.7 - 75.7i)T - 9.40e3iT^{2} \)
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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

−12.31898545809875219837155726441, −11.53314131832070149260518009017, −10.89092423433958233912549364926, −9.620283498419673905073278369577, −8.359415369466242936153390447686, −7.68937584309819821241499640795, −5.84591625970270956563864949156, −5.17605730920267577432342578317, −3.80551330695266203296631105707, −2.88474792796821257696299503368, 0.75806687225865809819225823030, 2.67677466503763542067090616273, 4.09308265974775899135853671338, 5.30664684243066333640747015734, 6.74583511811000700559206359097, 7.69347417907558282391389760473, 8.221455122924366391308168524391, 10.17291967748177576916357711263, 11.03854308295443314009839315599, 11.99322519800398757840998305926

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