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Properties

Label 2-810-9.4-c3-0-26
Degree $2$
Conductor $810$
Sign $0.342 - 0.939i$
Analytic cond. $47.7915$
Root an. cond. $6.91314$
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 + (−1.99 + 3.46i)4-s + (2.5 − 4.33i)5-s + (5.63 + 9.75i)7-s − 7.99·8-s + 10·10-s + (0.526 + 0.911i)11-s + (6.73 − 11.6i)13-s + (−11.2 + 19.5i)14-s + (−8 − 13.8i)16-s + 136.·17-s − 46.7·19-s + (10 + 17.3i)20-s + (−1.05 + 1.82i)22-s + (−10.9 + 18.9i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.223 − 0.387i)5-s + (0.304 + 0.526i)7-s − 0.353·8-s + 0.316·10-s + (0.0144 + 0.0249i)11-s + (0.143 − 0.248i)13-s + (−0.215 + 0.372i)14-s + (−0.125 − 0.216i)16-s + 1.95·17-s − 0.564·19-s + (0.111 + 0.193i)20-s + (−0.0102 + 0.0176i)22-s + (−0.0992 + 0.171i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $0.342 - 0.939i$
Analytic conductor: \(47.7915\)
Root analytic conductor: \(6.91314\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{810} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 810,\ (\ :3/2),\ 0.342 - 0.939i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.705461964\)
\(L(\frac12)\) \(\approx\) \(2.705461964\)
\(L(\frac{5}{2})\) 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 \( 1 \)
5 \( 1 + (-2.5 + 4.33i)T \)
good7 \( 1 + (-5.63 - 9.75i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-0.526 - 0.911i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-6.73 + 11.6i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 136.T + 4.91e3T^{2} \)
19 \( 1 + 46.7T + 6.85e3T^{2} \)
23 \( 1 + (10.9 - 18.9i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (54.1 + 93.7i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-28.5 + 49.4i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 223.T + 5.06e4T^{2} \)
41 \( 1 + (34.4 - 59.6i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-177. - 307. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-241. - 417. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 110.T + 1.48e5T^{2} \)
59 \( 1 + (328. - 569. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-19.2 - 33.3i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-339. + 587. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 572.T + 3.57e5T^{2} \)
73 \( 1 - 107.T + 3.89e5T^{2} \)
79 \( 1 + (-51.8 - 89.8i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-472. - 818. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 577.T + 7.04e5T^{2} \)
97 \( 1 + (-593. - 1.02e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−9.797658348908456489723316657561, −9.146351126453074516938748310039, −8.031935986624637377054366089741, −7.69911736930723468538336659327, −6.27093864395003992146646451887, −5.69100925684587731457847474122, −4.82923205793568458789127031290, −3.75211657987601716501074565985, −2.53501693111668176642457709162, −1.03195142390989393358185957079, 0.803313199201397178641016285336, 1.97015919960354392415525750369, 3.21627577360421039985220404794, 4.04559945472064871810612492962, 5.17964717221696286301883367144, 6.02150774584011956563649394421, 7.09900721521468538457229585937, 7.961340776269900822644769749401, 9.020278646394158419741481294573, 9.981627042960495391791376225582

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