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Properties

Label 2-210-105.47-c2-0-25
Degree $2$
Conductor $210$
Sign $-0.241 + 0.970i$
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  + (−0.366 − 1.36i)2-s + (0.874 − 2.86i)3-s + (−1.73 + i)4-s + (2.55 + 4.29i)5-s + (−4.24 − 0.144i)6-s + (6.45 − 2.70i)7-s + (2 + 1.99i)8-s + (−7.46 − 5.02i)9-s + (4.93 − 5.06i)10-s + (17.7 − 10.2i)11-s + (1.35 + 5.84i)12-s + (−10.2 − 10.2i)13-s + (−6.06 − 7.82i)14-s + (14.5 − 3.57i)15-s + (1.99 − 3.46i)16-s + (−4.39 + 16.4i)17-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (0.291 − 0.956i)3-s + (−0.433 + 0.250i)4-s + (0.511 + 0.859i)5-s + (−0.706 − 0.0241i)6-s + (0.922 − 0.386i)7-s + (0.250 + 0.249i)8-s + (−0.829 − 0.557i)9-s + (0.493 − 0.506i)10-s + (1.61 − 0.933i)11-s + (0.112 + 0.487i)12-s + (−0.790 − 0.790i)13-s + (−0.432 − 0.559i)14-s + (0.971 − 0.238i)15-s + (0.124 − 0.216i)16-s + (−0.258 + 0.965i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.241 + 0.970i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.241 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.241 + 0.970i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ -0.241 + 0.970i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.06235 - 1.35983i\)
\(L(\frac12)\) \(\approx\) \(1.06235 - 1.35983i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.366 + 1.36i)T \)
3 \( 1 + (-0.874 + 2.86i)T \)
5 \( 1 + (-2.55 - 4.29i)T \)
7 \( 1 + (-6.45 + 2.70i)T \)
good11 \( 1 + (-17.7 + 10.2i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (10.2 + 10.2i)T + 169iT^{2} \)
17 \( 1 + (4.39 - 16.4i)T + (-250. - 144.5i)T^{2} \)
19 \( 1 + (-11.5 + 20.0i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (20.5 - 5.50i)T + (458. - 264.5i)T^{2} \)
29 \( 1 + 2.30T + 841T^{2} \)
31 \( 1 + (-3.09 + 1.78i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-40.1 + 10.7i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 + 0.0268T + 1.68e3T^{2} \)
43 \( 1 + (6.00 - 6.00i)T - 1.84e3iT^{2} \)
47 \( 1 + (-11.8 + 3.18i)T + (1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (-2.41 + 8.99i)T + (-2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (13.7 - 7.94i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-41.9 - 24.2i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-6.37 + 23.7i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 - 80.4iT - 5.04e3T^{2} \)
73 \( 1 + (36.9 - 137. i)T + (-4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (-22.7 - 13.1i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (83.7 - 83.7i)T - 6.88e3iT^{2} \)
89 \( 1 + (39.3 + 22.7i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (84.5 - 84.5i)T - 9.40e3iT^{2} \)
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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

−11.58814096538243013704020296795, −11.22057540294684857573045458373, −9.973637879716215336942592964938, −8.874777478115215744534086986845, −7.88087594131500533649953992359, −6.85993245540146231892159144814, −5.72455035649924781029769558866, −3.81701022699018763030062739294, −2.49700727100548754790128053899, −1.16222247084228453697034666787, 1.85782594716327749246642273446, 4.27287937001613360778170600522, 4.85968568869353212648320189838, 6.05262277999825298589214197270, 7.54339585972271907897923861783, 8.673226864955739497044135969922, 9.419590792511955688971756690741, 9.919766688544639910706598501520, 11.60221162567183854763905177725, 12.20708904511664863566172699735

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