Properties

Label 2-210-21.2-c2-0-4
Degree $2$
Conductor $210$
Sign $-0.728 - 0.684i$
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  + (−1.22 − 0.707i)2-s + (1.92 + 2.29i)3-s + (0.999 + 1.73i)4-s + (−1.93 − 1.11i)5-s + (−0.732 − 4.17i)6-s + (−5.85 + 3.83i)7-s − 2.82i·8-s + (−1.57 + 8.86i)9-s + (1.58 + 2.73i)10-s + (−4.84 + 2.79i)11-s + (−2.05 + 5.63i)12-s − 1.42·13-s + (9.88 − 0.554i)14-s + (−1.15 − 6.60i)15-s + (−2.00 + 3.46i)16-s + (−14.6 + 8.47i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.642 + 0.766i)3-s + (0.249 + 0.433i)4-s + (−0.387 − 0.223i)5-s + (−0.122 − 0.696i)6-s + (−0.836 + 0.547i)7-s − 0.353i·8-s + (−0.175 + 0.984i)9-s + (0.158 + 0.273i)10-s + (−0.440 + 0.254i)11-s + (−0.171 + 0.469i)12-s − 0.109·13-s + (0.705 − 0.0396i)14-s + (−0.0772 − 0.440i)15-s + (−0.125 + 0.216i)16-s + (−0.863 + 0.498i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.728 - 0.684i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.728 - 0.684i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ -0.728 - 0.684i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.243089 + 0.613853i\)
\(L(\frac12)\) \(\approx\) \(0.243089 + 0.613853i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 + 0.707i)T \)
3 \( 1 + (-1.92 - 2.29i)T \)
5 \( 1 + (1.93 + 1.11i)T \)
7 \( 1 + (5.85 - 3.83i)T \)
good11 \( 1 + (4.84 - 2.79i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + 1.42T + 169T^{2} \)
17 \( 1 + (14.6 - 8.47i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-0.597 + 1.03i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (17.4 + 10.1i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 7.80iT - 841T^{2} \)
31 \( 1 + (-7.25 - 12.5i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (25.0 - 43.3i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 63.5iT - 1.68e3T^{2} \)
43 \( 1 - 70.4T + 1.84e3T^{2} \)
47 \( 1 + (24.0 + 13.8i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (57.6 - 33.3i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-85.4 + 49.3i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-55.7 + 96.6i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-11.6 - 20.1i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 15.8iT - 5.04e3T^{2} \)
73 \( 1 + (1.83 + 3.17i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-20.9 + 36.2i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 53.3iT - 6.88e3T^{2} \)
89 \( 1 + (-142. - 82.4i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 63.2T + 9.40e3T^{2} \)
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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

−12.49260673842168748392194388085, −11.34280652585115142072830077435, −10.33267046832460654125245986669, −9.579368331640453983750176125784, −8.684689818232926311935885007469, −7.924678311451371370353735157308, −6.48474077838762916360432638007, −4.84054103350422225028673915204, −3.54664002411204123206172872083, −2.35707311119905705310142890280, 0.39178036163603109793454635751, 2.43322282792777679955741579789, 3.85791049766876548751193068253, 5.90080092651356417459954740716, 6.99713303793435422222160125283, 7.59601696999184748364318214869, 8.695962561547509810813993167071, 9.595563215232325175007156152059, 10.65659245460465036401225051950, 11.78448042349864196463153890978

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