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Properties

Label 2-210-21.11-c2-0-11
Degree $2$
Conductor $210$
Sign $-0.930 + 0.366i$
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 + (−2.87 − 0.854i)3-s + (0.999 − 1.73i)4-s + (−1.93 + 1.11i)5-s + (4.12 − 0.987i)6-s + (6.85 − 1.41i)7-s + 2.82i·8-s + (7.54 + 4.91i)9-s + (1.58 − 2.73i)10-s + (1.73 + 1.00i)11-s + (−4.35 + 4.12i)12-s − 17.1·13-s + (−7.39 + 6.58i)14-s + (6.52 − 1.56i)15-s + (−2.00 − 3.46i)16-s + (−16.4 − 9.47i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.958 − 0.284i)3-s + (0.249 − 0.433i)4-s + (−0.387 + 0.223i)5-s + (0.687 − 0.164i)6-s + (0.979 − 0.202i)7-s + 0.353i·8-s + (0.837 + 0.545i)9-s + (0.158 − 0.273i)10-s + (0.157 + 0.0911i)11-s + (−0.362 + 0.343i)12-s − 1.31·13-s + (−0.528 + 0.470i)14-s + (0.434 − 0.104i)15-s + (−0.125 − 0.216i)16-s + (−0.965 − 0.557i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.930 + 0.366i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.930 + 0.366i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.930 + 0.366i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ -0.930 + 0.366i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0157721 - 0.0830936i\)
\(L(\frac12)\) \(\approx\) \(0.0157721 - 0.0830936i\)
\(L(2)\) 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 \( 1 + (2.87 + 0.854i)T \)
5 \( 1 + (1.93 - 1.11i)T \)
7 \( 1 + (-6.85 + 1.41i)T \)
good11 \( 1 + (-1.73 - 1.00i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + 17.1T + 169T^{2} \)
17 \( 1 + (16.4 + 9.47i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (7.18 + 12.4i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (22.4 - 12.9i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + 10.3iT - 841T^{2} \)
31 \( 1 + (24.0 - 41.5i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (21.9 + 38.0i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 22.8iT - 1.68e3T^{2} \)
43 \( 1 + 73.6T + 1.84e3T^{2} \)
47 \( 1 + (20.0 - 11.5i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-45.3 - 26.1i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-43.9 - 25.3i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (38.9 + 67.4i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (13.9 - 24.0i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 50.6iT - 5.04e3T^{2} \)
73 \( 1 + (35.5 - 61.5i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-17.4 - 30.1i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 131. iT - 6.88e3T^{2} \)
89 \( 1 + (45.4 - 26.2i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 71.0T + 9.40e3T^{2} \)
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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

−11.55931089325459821672099830846, −10.83903439833098964775323096774, −9.891699197913114794563559033819, −8.580654772709970619787134478339, −7.38039770500079007986274901092, −6.91634359434071473700635494107, −5.40890514795349930642407159458, −4.48433268431469586636417583098, −1.97132573009339275383393625544, −0.06254472159006642700309195565, 1.88191556127438425609521364197, 4.07570774871947945629247291093, 5.06615137511329466641241471076, 6.46714417974493967792137743820, 7.68220314410149997472872802526, 8.637898471735848050081397351165, 9.861566299763705863960163265796, 10.65891109892976099140648181212, 11.65589159108211825400991636774, 12.07176558291826162478157630882

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