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Properties

Label 2-210-35.9-c1-0-2
Degree $2$
Conductor $210$
Sign $0.982 + 0.185i$
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  + (−0.866 − 0.5i)2-s + (0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s + (−0.133 + 2.23i)5-s − 0.999·6-s + (1.73 − 2i)7-s − 0.999i·8-s + (0.499 − 0.866i)9-s + (1.23 − 1.86i)10-s + (2.5 + 4.33i)11-s + (0.866 + 0.499i)12-s + i·13-s + (−2.5 + 0.866i)14-s + (1 + 1.99i)15-s + (−0.5 + 0.866i)16-s + (1.73 − i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.499 − 0.288i)3-s + (0.249 + 0.433i)4-s + (−0.0599 + 0.998i)5-s − 0.408·6-s + (0.654 − 0.755i)7-s − 0.353i·8-s + (0.166 − 0.288i)9-s + (0.389 − 0.590i)10-s + (0.753 + 1.30i)11-s + (0.249 + 0.144i)12-s + 0.277i·13-s + (−0.668 + 0.231i)14-s + (0.258 + 0.516i)15-s + (−0.125 + 0.216i)16-s + (0.420 − 0.242i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.185i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 + 0.185i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.982 + 0.185i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ 0.982 + 0.185i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14118 - 0.106840i\)
\(L(\frac12)\) \(\approx\) \(1.14118 - 0.106840i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 + (0.133 - 2.23i)T \)
7 \( 1 + (-1.73 + 2i)T \)
good11 \( 1 + (-2.5 - 4.33i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - iT - 13T^{2} \)
17 \( 1 + (-1.73 + i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.5 + 6.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.59 + 1.5i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (-3 - 5.19i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.33 + 2.5i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 9T + 41T^{2} \)
43 \( 1 - 10iT - 43T^{2} \)
47 \( 1 + (11.2 + 6.5i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.866 - 0.5i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.19 - 3i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 + (-3.46 + 2i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (7 - 12.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 10iT - 83T^{2} \)
89 \( 1 + (-5 + 8.66i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 8iT - 97T^{2} \)
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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

−12.02235753705840457009937498452, −11.39117332816808950014467798564, −10.24960616383089923338695279341, −9.584904743817233895065080360825, −8.313293496572333223869127736661, −7.19436015372326548269831519251, −6.83579701117063113614557987644, −4.57687865075359049495786459565, −3.20795553169858875821681464774, −1.73328792566260583794607486458, 1.53774337077871594567512854657, 3.59722019306561726796420886424, 5.20261531446482925378923494495, 6.05064931187087773186773629645, 7.940106893322321348302467614680, 8.359264956592400458377634175427, 9.223592974105919773077097800917, 10.14480761795054286053445159542, 11.53274759513886522619930169779, 12.14462747383253310133999046703

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