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Properties

Label 2-210-7.2-c3-0-10
Degree $2$
Conductor $210$
Sign $0.146 + 0.989i$
Analytic cond. $12.3904$
Root an. cond. $3.52000$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + 1.73i)2-s + (1.5 + 2.59i)3-s + (−1.99 − 3.46i)4-s + (−2.5 + 4.33i)5-s − 6·6-s + (−18.4 + 1.54i)7-s + 7.99·8-s + (−4.5 + 7.79i)9-s + (−5 − 8.66i)10-s + (−11.9 − 20.7i)11-s + (6.00 − 10.3i)12-s + 27.7·13-s + (15.7 − 33.5i)14-s − 15.0·15-s + (−8 + 13.8i)16-s + (−56.3 − 97.5i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.223 + 0.387i)5-s − 0.408·6-s + (−0.996 + 0.0833i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.158 − 0.273i)10-s + (−0.327 − 0.567i)11-s + (0.144 − 0.249i)12-s + 0.592·13-s + (0.301 − 0.639i)14-s − 0.258·15-s + (−0.125 + 0.216i)16-s + (−0.803 − 1.39i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.146 + 0.989i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.146 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.146 + 0.989i$
Analytic conductor: \(12.3904\)
Root analytic conductor: \(3.52000\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :3/2),\ 0.146 + 0.989i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.255613 - 0.220593i\)
\(L(\frac12)\) \(\approx\) \(0.255613 - 0.220593i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - 1.73i)T \)
3 \( 1 + (-1.5 - 2.59i)T \)
5 \( 1 + (2.5 - 4.33i)T \)
7 \( 1 + (18.4 - 1.54i)T \)
good11 \( 1 + (11.9 + 20.7i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 27.7T + 2.19e3T^{2} \)
17 \( 1 + (56.3 + 97.5i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (15.7 - 27.2i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-26.3 + 45.5i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 113.T + 2.43e4T^{2} \)
31 \( 1 + (42.6 + 73.8i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-194. + 337. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 165.T + 6.89e4T^{2} \)
43 \( 1 - 116.T + 7.95e4T^{2} \)
47 \( 1 + (48.9 - 84.7i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (173. + 300. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (87.1 + 150. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (266. - 462. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (187. + 324. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 739.T + 3.57e5T^{2} \)
73 \( 1 + (12.1 + 21.1i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (594. - 1.02e3i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 565.T + 5.71e5T^{2} \)
89 \( 1 + (486. - 842. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 80.5T + 9.12e5T^{2} \)
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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

−11.40534354451918802323949154048, −10.59511489490476494071015107655, −9.513854861619220846725210611060, −8.847685646909448313329951103942, −7.64839958205491263128134531092, −6.60523565514622552144915367937, −5.56163101012144150681456424135, −4.05006607774036186080986562464, −2.75945874214324766402009819211, −0.15312367793849538662497046440, 1.58891008892998587267561310216, 3.10038216938883727089728242335, 4.31478978103139535836459832802, 6.08144227878196622903871373209, 7.20651636696616838248019539442, 8.350691317718285726194379990170, 9.146713522778855478342899994676, 10.16854031067427252606883435640, 11.16136441961169746756762841307, 12.28305982253697306078601969513

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