Properties

Degree $2$
Conductor $210$
Sign $0.605 - 0.795i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (−0.5 + 0.866i)5-s + 0.999·6-s + (2 + 1.73i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (−0.499 − 0.866i)10-s + (0.5 + 0.866i)11-s + (−0.499 + 0.866i)12-s + 7·13-s + (−2.5 + 0.866i)14-s + 0.999·15-s + (−0.5 + 0.866i)16-s + (2 + 3.46i)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 + (0.755 + 0.654i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.158 − 0.273i)10-s + (0.150 + 0.261i)11-s + (−0.144 + 0.249i)12-s + 1.94·13-s + (−0.668 + 0.231i)14-s + 0.258·15-s + (−0.125 + 0.216i)16-s + (0.485 + 0.840i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.605 - 0.795i$
Motivic weight: \(1\)
Character: $\chi_{210} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ 0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.872295 + 0.432420i\)
\(L(\frac12)\) \(\approx\) \(0.872295 + 0.432420i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-2 - 1.73i)T \)
good11 \( 1 + (-0.5 - 0.866i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 7T + 13T^{2} \)
17 \( 1 + (-2 - 3.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 8T + 29T^{2} \)
31 \( 1 + (3 + 5.19i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.5 + 2.59i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 9T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (-1.5 + 2.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6 + 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2 + 3.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6 + 10.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 14T + 71T^{2} \)
73 \( 1 + (-7 - 12.1i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2 - 3.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + (-1 + 1.73i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 16T + 97T^{2} \)
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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

−12.54222679626572656058241459616, −11.31436905617365287546044175032, −10.82982075231035922705162496598, −9.321348377552733244967478787206, −8.301127107896027728664376862286, −7.60354183261529834616723244201, −6.25500043333902438535148609731, −5.61158003553484384411560577567, −3.90167171031770626257004290166, −1.69377058548200865114807002463, 1.19813431854546906531279411642, 3.48712101176861286367841274171, 4.46917981207639980008355754257, 5.78941777267089190256945797876, 7.39747376377291974433566265062, 8.500901659901231653412929222067, 9.265336235159572918269579666828, 10.57486391943161221813996213901, 11.12932067639503526792264648730, 11.87709362304223388791814807686

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