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Properties

Label 2-210-21.5-c1-0-1
Degree $2$
Conductor $210$
Sign $-0.181 - 0.983i$
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 + (−0.511 + 1.65i)3-s + (0.499 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (1.27 − 1.17i)6-s + (2.63 − 0.226i)7-s − 0.999i·8-s + (−2.47 − 1.69i)9-s + (0.866 − 0.499i)10-s + (−4.29 + 2.47i)11-s + (−1.68 + 0.384i)12-s + 6.37i·13-s + (−2.39 − 1.12i)14-s + (−1.17 − 1.27i)15-s + (−0.5 + 0.866i)16-s + (1.81 + 3.14i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.295 + 0.955i)3-s + (0.249 + 0.433i)4-s + (−0.223 + 0.387i)5-s + (0.518 − 0.480i)6-s + (0.996 − 0.0854i)7-s − 0.353i·8-s + (−0.825 − 0.563i)9-s + (0.273 − 0.158i)10-s + (−1.29 + 0.747i)11-s + (−0.487 + 0.111i)12-s + 1.76i·13-s + (−0.640 − 0.299i)14-s + (−0.304 − 0.327i)15-s + (−0.125 + 0.216i)16-s + (0.439 + 0.761i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.181 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.181 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.181 - 0.983i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ -0.181 - 0.983i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.453636 + 0.545229i\)
\(L(\frac12)\) \(\approx\) \(0.453636 + 0.545229i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.511 - 1.65i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-2.63 + 0.226i)T \)
good11 \( 1 + (4.29 - 2.47i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 6.37iT - 13T^{2} \)
17 \( 1 + (-1.81 - 3.14i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.64 + 2.10i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.65 - 0.958i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 0.800iT - 29T^{2} \)
31 \( 1 + (-4.36 + 2.51i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.91 - 5.04i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 7.45T + 41T^{2} \)
43 \( 1 + 4.64T + 43T^{2} \)
47 \( 1 + (-0.303 + 0.526i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-12.3 + 7.15i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.875 + 1.51i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.96 - 1.71i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.99 + 6.91i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.48iT - 71T^{2} \)
73 \( 1 + (-13.0 + 7.54i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.33 - 2.31i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 7.27T + 83T^{2} \)
89 \( 1 + (-0.942 + 1.63i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 17.6iT - 97T^{2} \)
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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

−12.18738894758312757796094787612, −11.39312682574128767475886446950, −10.68043359974153703867283160738, −9.915521689745838707610124221096, −8.806600369300413880071405093550, −7.86719873266951100383477646336, −6.61060454450803970852635843204, −4.98418012360482858868921064384, −4.04044082430598872472251021924, −2.27116085195423755774582030279, 0.77153059647359680411306904137, 2.65013800700032133634406644616, 5.16734103322245316567322461930, 5.76552483341579440323572324622, 7.37487263412713806117815327461, 8.069676170624039781358861239972, 8.593934987105557651862087135657, 10.41427075151402359550321318322, 10.98118553933143693602149516613, 12.12881044623141903439974184626

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