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Properties

Label 2-210-21.5-c1-0-5
Degree $2$
Conductor $210$
Sign $0.966 - 0.255i$
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 + (−1.68 − 0.384i)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 + (2.70 + 1.29i)9-s + (0.866 − 0.499i)10-s + (4.29 − 2.47i)11-s + (−0.511 − 1.65i)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 + (−0.974 − 0.222i)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 + (0.901 + 0.433i)9-s + (0.273 − 0.158i)10-s + (1.29 − 0.747i)11-s + (−0.147 − 0.477i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.255i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 - 0.255i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.966 - 0.255i$
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: \((2,\ 210,\ (\ :1/2),\ 0.966 - 0.255i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.44223 + 0.187312i\)
\(L(\frac12)\) \(\approx\) \(1.44223 + 0.187312i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.68 + 0.384i)T \)
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

−12.13542935255028760853762953365, −11.65774137068437995113526553019, −10.95755479649844559926979985086, −9.359482602798001309347181574773, −8.335351408617369398834502778842, −6.85565564686052563715218667291, −6.30613500330582412911271193031, −4.89235208054753378019440643683, −4.24833499223080492881474288933, −1.69975663870794962268082063571, 1.69465962564745372882092565780, 3.78418298920647880296727136416, 4.88779586122971781622466634799, 5.91727547635574325456429691523, 6.86917882077629347986294574208, 8.292992868535625015414004412656, 9.902825599854247821492861771143, 10.57247912922124433573490370326, 11.40806406326477414853079798686, 12.26438359740099532985390440402

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