Properties

Label 2-810-9.7-c3-0-44
Degree $2$
Conductor $810$
Sign $-0.766 - 0.642i$
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 + (21 − 36.3i)11-s + (−10 − 17.3i)13-s + (−3.99 − 6.92i)14-s + (−8 + 13.8i)16-s − 93·17-s + 59·19-s + (−10 + 17.3i)20-s + (−42 − 72.7i)22-s + (4.5 + 7.79i)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 + (0.575 − 0.996i)11-s + (−0.213 − 0.369i)13-s + (−0.0763 − 0.132i)14-s + (−0.125 + 0.216i)16-s − 1.32·17-s + 0.712·19-s + (−0.111 + 0.193i)20-s + (−0.407 − 0.704i)22-s + (0.0407 + 0.0706i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $-0.766 - 0.642i$
Analytic conductor: \(47.7915\)
Root analytic conductor: \(6.91314\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{810} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 810,\ (\ :3/2),\ -0.766 - 0.642i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9240685866\)
\(L(\frac12)\) \(\approx\) \(0.9240685866\)
\(L(\frac{5}{2})\) 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 \( 1 + (-21 + 36.3i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (10 + 17.3i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 93T + 4.91e3T^{2} \)
19 \( 1 - 59T + 6.85e3T^{2} \)
23 \( 1 + (-4.5 - 7.79i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-60 + 103. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (23.5 + 40.7i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 262T + 5.06e4T^{2} \)
41 \( 1 + (-63 - 109. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-89 + 154. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-72 + 124. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 741T + 1.48e5T^{2} \)
59 \( 1 + (222 + 384. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (110.5 - 191. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-269 - 465. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 690T + 3.57e5T^{2} \)
73 \( 1 + 1.12e3T + 3.89e5T^{2} \)
79 \( 1 + (332.5 - 575. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-37.5 + 64.9i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 1.08e3T + 7.04e5T^{2} \)
97 \( 1 + (772 - 1.33e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−9.305636941230849243106292939456, −8.703087283481579300676929845677, −7.71849758412349843201552210072, −6.57794196636827774390423265585, −5.65332578864454022220749837141, −4.65785461220159410841570971472, −3.79351773242808896595876118518, −2.75073584178580522709338437894, −1.37046784436751794002850280141, −0.21669737127033303428434861931, 1.78585113414510919051822459903, 3.07844079670064303598111443721, 4.26648962104952081694893654842, 4.93664643501540634929347995162, 6.17027099838268383320407887343, 6.96254969412827527707059194670, 7.51465468573401674226038709066, 8.716974368978870460528275916347, 9.310517363530664891794130067158, 10.35244089591395288850059947338

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