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Properties

Label 2-210-105.2-c1-0-14
Degree $2$
Conductor $210$
Sign $0.793 + 0.608i$
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  + (0.258 + 0.965i)2-s + (0.365 − 1.69i)3-s + (−0.866 + 0.499i)4-s + (1.50 − 1.65i)5-s + (1.72 − 0.0855i)6-s + (−1.18 − 2.36i)7-s + (−0.707 − 0.707i)8-s + (−2.73 − 1.23i)9-s + (1.98 + 1.02i)10-s + (2.97 − 1.71i)11-s + (0.530 + 1.64i)12-s + (−3.87 + 3.87i)13-s + (1.97 − 1.75i)14-s + (−2.25 − 3.15i)15-s + (0.500 − 0.866i)16-s + (3.70 + 0.993i)17-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (0.210 − 0.977i)3-s + (−0.433 + 0.249i)4-s + (0.672 − 0.739i)5-s + (0.706 − 0.0349i)6-s + (−0.448 − 0.893i)7-s + (−0.249 − 0.249i)8-s + (−0.911 − 0.412i)9-s + (0.628 + 0.324i)10-s + (0.895 − 0.517i)11-s + (0.153 + 0.475i)12-s + (−1.07 + 1.07i)13-s + (0.528 − 0.469i)14-s + (−0.581 − 0.813i)15-s + (0.125 − 0.216i)16-s + (0.899 + 0.240i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.793 + 0.608i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ 0.793 + 0.608i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32146 - 0.447939i\)
\(L(\frac12)\) \(\approx\) \(1.32146 - 0.447939i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 - 0.965i)T \)
3 \( 1 + (-0.365 + 1.69i)T \)
5 \( 1 + (-1.50 + 1.65i)T \)
7 \( 1 + (1.18 + 2.36i)T \)
good11 \( 1 + (-2.97 + 1.71i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.87 - 3.87i)T - 13iT^{2} \)
17 \( 1 + (-3.70 - 0.993i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-5.97 - 3.44i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.11 + 0.298i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 2.35T + 29T^{2} \)
31 \( 1 + (-3.58 - 6.20i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.77 - 1.28i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 7.73iT - 41T^{2} \)
43 \( 1 + (0.576 - 0.576i)T - 43iT^{2} \)
47 \( 1 + (2.28 + 8.51i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.246 + 0.921i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (2.74 + 4.75i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.442 + 0.766i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.15 + 4.32i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 1.76iT - 71T^{2} \)
73 \( 1 + (6.82 + 1.83i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-4.97 - 2.87i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.09 - 7.09i)T + 83iT^{2} \)
89 \( 1 + (0.641 - 1.11i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.15 - 2.15i)T + 97iT^{2} \)
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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

−12.39703569947484230829669449583, −11.79248386484623019956563335560, −9.941634893787277696610948751246, −9.199255640765595824094972577741, −8.083390062099449138570468134416, −7.05988871543255665082191268479, −6.27925968625229875111146046500, −5.07553214336450796176850206850, −3.47962719008476013446315157062, −1.31825949087443456772302388676, 2.54114467583185759037928818991, 3.36205731500651638481638711618, 5.04436189999715885000054582497, 5.84998752009552640965004101227, 7.46158570306746068941320581736, 9.141052175804902213342081449583, 9.659399189982567850944136514193, 10.31712409581665590549958630025, 11.50451816257125526437668079725, 12.25178246230116989981224986605

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