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Properties

Label 2-810-9.4-c3-0-19
Degree $2$
Conductor $810$
Sign $-0.173 - 0.984i$
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 + (2 + 3.46i)7-s − 7.99·8-s − 10·10-s + (−24 − 41.5i)11-s + (−1 + 1.73i)13-s + (−3.99 + 6.92i)14-s + (−8 − 13.8i)16-s + 114·17-s + 140·19-s + (−10 − 17.3i)20-s + (48 − 83.1i)22-s + (36 − 62.3i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.223 + 0.387i)5-s + (0.107 + 0.187i)7-s − 0.353·8-s − 0.316·10-s + (−0.657 − 1.13i)11-s + (−0.0213 + 0.0369i)13-s + (−0.0763 + 0.132i)14-s + (−0.125 − 0.216i)16-s + 1.62·17-s + 1.69·19-s + (−0.111 − 0.193i)20-s + (0.465 − 0.805i)22-s + (0.326 − 0.565i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $-0.173 - 0.984i$
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: \((2,\ 810,\ (\ :3/2),\ -0.173 - 0.984i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.206869681\)
\(L(\frac12)\) \(\approx\) \(2.206869681\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2 - 3.46i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (24 + 41.5i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 114T + 4.91e3T^{2} \)
19 \( 1 - 140T + 6.85e3T^{2} \)
23 \( 1 + (-36 + 62.3i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-105 - 181. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (136 - 235. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 334T + 5.06e4T^{2} \)
41 \( 1 + (99 - 171. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-134 - 232. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-108 - 187. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 78T + 1.48e5T^{2} \)
59 \( 1 + (-120 + 207. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (151 + 261. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (298 - 516. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 768T + 3.57e5T^{2} \)
73 \( 1 + 478T + 3.89e5T^{2} \)
79 \( 1 + (-320 - 554. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (174 + 301. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 210T + 7.04e5T^{2} \)
97 \( 1 + (-767 - 1.32e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−10.15654237602487405573337490213, −9.042345667320217032631065944274, −8.224241849301056216918337304864, −7.49682792104555461771904144127, −6.66862086101920496331719031501, −5.49414733823382487935011348593, −5.12196319973062219200578936599, −3.45053147291626498129051716096, −3.02986739073575827442992807115, −1.03960538447744075143387245361, 0.64242935669352938397950764797, 1.83626307391308889629290670794, 3.11130975132747923062393873450, 4.07213867728197129802241045458, 5.15419101051553368630762217472, 5.67861809605018058164066081574, 7.34696231272868379192233176262, 7.69031941439357464882675871816, 8.994855848758337873062899141920, 9.880599521252069768472748533219

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