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Properties

Label 2-210-35.3-c1-0-6
Degree $2$
Conductor $210$
Sign $-0.658 + 0.752i$
Analytic cond. $1.67685$
Root an. cond. $1.29493$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.965 − 0.258i)2-s + (−0.258 − 0.965i)3-s + (0.866 + 0.499i)4-s + (0.519 − 2.17i)5-s + i·6-s + (−2.64 − 0.153i)7-s + (−0.707 − 0.707i)8-s + (−0.866 + 0.499i)9-s + (−1.06 + 1.96i)10-s + (2.27 − 3.94i)11-s + (0.258 − 0.965i)12-s + (−1.77 + 1.77i)13-s + (2.51 + 0.831i)14-s + (−2.23 + 0.0614i)15-s + (0.500 + 0.866i)16-s + (−3.98 + 1.06i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (−0.149 − 0.557i)3-s + (0.433 + 0.249i)4-s + (0.232 − 0.972i)5-s + 0.408i·6-s + (−0.998 − 0.0579i)7-s + (−0.249 − 0.249i)8-s + (−0.288 + 0.166i)9-s + (−0.336 + 0.621i)10-s + (0.686 − 1.18i)11-s + (0.0747 − 0.278i)12-s + (−0.493 + 0.493i)13-s + (0.671 + 0.222i)14-s + (−0.577 + 0.0158i)15-s + (0.125 + 0.216i)16-s + (−0.966 + 0.258i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.658 + 0.752i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.658 + 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.658 + 0.752i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ -0.658 + 0.752i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.273922 - 0.603853i\)
\(L(\frac12)\) \(\approx\) \(0.273922 - 0.603853i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 + 0.258i)T \)
3 \( 1 + (0.258 + 0.965i)T \)
5 \( 1 + (-0.519 + 2.17i)T \)
7 \( 1 + (2.64 + 0.153i)T \)
good11 \( 1 + (-2.27 + 3.94i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.77 - 1.77i)T - 13iT^{2} \)
17 \( 1 + (3.98 - 1.06i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.88 + 3.27i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.08 + 7.77i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + 1.55iT - 29T^{2} \)
31 \( 1 + (-3.37 - 1.94i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-11.0 - 2.95i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 11.3iT - 41T^{2} \)
43 \( 1 + (-0.367 - 0.367i)T + 43iT^{2} \)
47 \( 1 + (-1.30 + 4.87i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-8.14 + 2.18i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-0.221 + 0.383i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.09 + 4.09i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.41 - 8.99i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 6.68T + 71T^{2} \)
73 \( 1 + (1.12 + 4.20i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-4.08 + 2.35i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.21 + 3.21i)T - 83iT^{2} \)
89 \( 1 + (3.02 + 5.23i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.462 + 0.462i)T + 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.97545027739261735379645353854, −11.13980495997872790986531322209, −9.882539181352451575647446452562, −8.927146942544445699782920361818, −8.363268510774512988024958379843, −6.75688071681239070625704163474, −6.16456609186569460476980198853, −4.41231811602834028382547899510, −2.59055967443050078886782986182, −0.70447370768372466001486010477, 2.46662526222410467302949865496, 3.93700398740594616180391216593, 5.70292495705456051539511309506, 6.73131767740738759006806660090, 7.51274421245381102966778754697, 9.195959097528946705198227476751, 9.745788893660121016453976499066, 10.50374007457171703394688262211, 11.51001729600862474201824412272, 12.56165567936761573320540579869

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