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Properties

Label 2-810-9.4-c3-0-28
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 + (4.90 + 8.50i)7-s − 7.99·8-s − 10·10-s + (9.53 + 16.5i)11-s + (43.7 − 75.7i)13-s + (−9.81 + 17.0i)14-s + (−8 − 13.8i)16-s + 10.8·17-s + 103.·19-s + (−10 − 17.3i)20-s + (−19.0 + 33.0i)22-s + (98.0 − 169. i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.223 + 0.387i)5-s + (0.265 + 0.459i)7-s − 0.353·8-s − 0.316·10-s + (0.261 + 0.452i)11-s + (0.933 − 1.61i)13-s + (−0.187 + 0.324i)14-s + (−0.125 − 0.216i)16-s + 0.154·17-s + 1.24·19-s + (−0.111 − 0.193i)20-s + (−0.184 + 0.320i)22-s + (0.889 − 1.54i)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

\(L(2)\) \(\approx\) \(2.643412404\)
\(L(\frac12)\) \(\approx\) \(2.643412404\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.90 - 8.50i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-9.53 - 16.5i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-43.7 + 75.7i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 10.8T + 4.91e3T^{2} \)
19 \( 1 - 103.T + 6.85e3T^{2} \)
23 \( 1 + (-98.0 + 169. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-6.04 - 10.4i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (117. - 203. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 304.T + 5.06e4T^{2} \)
41 \( 1 + (182. - 315. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (169. + 294. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-118. - 205. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 31.2T + 1.48e5T^{2} \)
59 \( 1 + (64.8 - 112. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-269. - 466. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (168. - 291. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 264.T + 3.57e5T^{2} \)
73 \( 1 - 950.T + 3.89e5T^{2} \)
79 \( 1 + (187. + 324. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-538. - 932. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 909.T + 7.04e5T^{2} \)
97 \( 1 + (658. + 1.14e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−10.06838484047730522731735237514, −8.916015433364268592301054489661, −8.233043597506480564408682085543, −7.41189912306661578404876741978, −6.53720304151455673237022042628, −5.59473421520727955237738494979, −4.85150579458101115799273644785, −3.56504166000998651951184776091, −2.75024568170814441693363046724, −0.947204349678283651579676512680, 0.880510159846202176790458016165, 1.77171098404364519085432747222, 3.39103474266844209709751699839, 4.04922749972395289524343149244, 5.07129090445352361947335348480, 6.02563146449339819308795792654, 7.10748209901634972737975475485, 8.029533976855552437617364752223, 9.230412961459057401294241101542, 9.461807142596247391589427126913

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