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Properties

Label 2-210-105.17-c2-0-20
Degree $2$
Conductor $210$
Sign $0.498 - 0.866i$
Analytic cond. $5.72208$
Root an. cond. $2.39208$
Motivic weight $2$
Arithmetic yes
Rational no
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 + (1.97 + 2.25i)3-s + (1.73 + i)4-s + (3.75 + 3.29i)5-s + (1.87 + 3.80i)6-s + (5.65 − 4.12i)7-s + (1.99 + 2i)8-s + (−1.16 + 8.92i)9-s + (3.92 + 5.87i)10-s + (−10.7 − 6.20i)11-s + (1.17 + 5.88i)12-s + (−17.5 − 17.5i)13-s + (9.23 − 3.55i)14-s + (0.0154 + 14.9i)15-s + (1.99 + 3.46i)16-s + (9.60 − 2.57i)17-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (0.659 + 0.751i)3-s + (0.433 + 0.250i)4-s + (0.751 + 0.659i)5-s + (0.313 + 0.633i)6-s + (0.808 − 0.588i)7-s + (0.249 + 0.250i)8-s + (−0.128 + 0.991i)9-s + (0.392 + 0.587i)10-s + (−0.976 − 0.563i)11-s + (0.0979 + 0.490i)12-s + (−1.34 − 1.34i)13-s + (0.659 − 0.254i)14-s + (0.00102 + 0.999i)15-s + (0.124 + 0.216i)16-s + (0.565 − 0.151i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.498 - 0.866i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.498 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.498 - 0.866i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ 0.498 - 0.866i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.63724 + 1.52557i\)
\(L(\frac12)\) \(\approx\) \(2.63724 + 1.52557i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 - 0.366i)T \)
3 \( 1 + (-1.97 - 2.25i)T \)
5 \( 1 + (-3.75 - 3.29i)T \)
7 \( 1 + (-5.65 + 4.12i)T \)
good11 \( 1 + (10.7 + 6.20i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (17.5 + 17.5i)T + 169iT^{2} \)
17 \( 1 + (-9.60 + 2.57i)T + (250. - 144.5i)T^{2} \)
19 \( 1 + (-6.00 - 10.3i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (0.588 - 2.19i)T + (-458. - 264.5i)T^{2} \)
29 \( 1 - 19.7T + 841T^{2} \)
31 \( 1 + (45.3 + 26.2i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (1.97 - 7.35i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 - 18.4T + 1.68e3T^{2} \)
43 \( 1 + (-33.8 + 33.8i)T - 1.84e3iT^{2} \)
47 \( 1 + (6.09 - 22.7i)T + (-1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (76.5 - 20.5i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (6.05 + 3.49i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (5.55 - 3.20i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-74.1 + 19.8i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + 29.3iT - 5.04e3T^{2} \)
73 \( 1 + (-14.2 + 3.82i)T + (4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (-118. + 68.4i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (12.9 - 12.9i)T - 6.88e3iT^{2} \)
89 \( 1 + (91.0 - 52.5i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (69.6 - 69.6i)T - 9.40e3iT^{2} \)
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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

−12.56495628784863958167905623442, −11.01059314013260598403942245971, −10.46989445246205635006028731340, −9.612042495857026979047468973067, −7.946480379780223907418397469239, −7.50938034498705663114143755974, −5.64394357870959596235806001398, −4.98470091778908994431152093696, −3.42149537553450972176314400729, −2.41031537998144118001897897174, 1.74013489970346591989229842826, 2.58260079682961993742523309884, 4.61704484923058819024999550072, 5.46700052569655081204594922896, 6.86907753171989776902803429177, 7.86344262715819796896226831456, 9.048131490056859496660124240434, 9.849436837158141297451498976845, 11.35606481126108490987652841346, 12.52370576218702917797091495858

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