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Properties

Label 2-210-21.2-c2-0-11
Degree $2$
Conductor $210$
Sign $0.946 - 0.322i$
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 + (−1.44 − 2.63i)3-s + (0.999 + 1.73i)4-s + (1.93 + 1.11i)5-s + (0.0971 − 4.24i)6-s + (1.98 + 6.71i)7-s + 2.82i·8-s + (−4.85 + 7.58i)9-s + (1.58 + 2.73i)10-s + (11.3 − 6.55i)11-s + (3.11 − 5.12i)12-s + 20.0·13-s + (−2.31 + 9.62i)14-s + (0.153 − 6.70i)15-s + (−2.00 + 3.46i)16-s + (7.44 − 4.29i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.480 − 0.877i)3-s + (0.249 + 0.433i)4-s + (0.387 + 0.223i)5-s + (0.0161 − 0.706i)6-s + (0.283 + 0.958i)7-s + 0.353i·8-s + (−0.539 + 0.842i)9-s + (0.158 + 0.273i)10-s + (1.03 − 0.595i)11-s + (0.259 − 0.427i)12-s + 1.53·13-s + (−0.165 + 0.687i)14-s + (0.0102 − 0.447i)15-s + (−0.125 + 0.216i)16-s + (0.437 − 0.252i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 - 0.322i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.946 - 0.322i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.946 - 0.322i$
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: \((2,\ 210,\ (\ :1),\ 0.946 - 0.322i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.10713 + 0.348953i\)
\(L(\frac12)\) \(\approx\) \(2.10713 + 0.348953i\)
\(L(2)\) 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 \( 1 + (1.44 + 2.63i)T \)
5 \( 1 + (-1.93 - 1.11i)T \)
7 \( 1 + (-1.98 - 6.71i)T \)
good11 \( 1 + (-11.3 + 6.55i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 - 20.0T + 169T^{2} \)
17 \( 1 + (-7.44 + 4.29i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-3.25 + 5.63i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (33.1 + 19.1i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 23.6iT - 841T^{2} \)
31 \( 1 + (-22.0 - 38.2i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (9.37 - 16.2i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 14.7iT - 1.68e3T^{2} \)
43 \( 1 + 36.8T + 1.84e3T^{2} \)
47 \( 1 + (69.0 + 39.8i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (22.0 - 12.7i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-87.3 + 50.4i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (24.7 - 42.8i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (29.3 + 50.7i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 82.7iT - 5.04e3T^{2} \)
73 \( 1 + (34.8 + 60.2i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (0.932 - 1.61i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 123. iT - 6.88e3T^{2} \)
89 \( 1 + (-53.2 - 30.7i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 113.T + 9.40e3T^{2} \)
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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

−12.08733266723382647001700108362, −11.65483155719356016689107578025, −10.57566599361259592982667236551, −8.839698085016177475619308279621, −8.162608775805713319926301550516, −6.58598334378988785291140788807, −6.16531901778672807625040504219, −5.09421996980146132048759835163, −3.27788053450025397813089610729, −1.63422776601355304465235360850, 1.35101416763504484285953705359, 3.72555334333461419258990105561, 4.28456089871689116751288859427, 5.70469720203800890146836107564, 6.51419570181646003081438574298, 8.175483833637924358363591632744, 9.633406753285726389016728416370, 10.13189951191609096617846310666, 11.30056003201927750120394935218, 11.79867050793923105366246830386

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