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Properties

Label 2-810-9.7-c3-0-42
Degree $2$
Conductor $810$
Sign $-0.939 + 0.342i$
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 + (14.7 − 25.5i)7-s − 7.99·8-s + 10·10-s + (23.2 − 40.3i)11-s + (−46.0 − 79.7i)13-s + (−29.5 − 51.1i)14-s + (−8 + 13.8i)16-s + 4.53·17-s + 87.5·19-s + (10 − 17.3i)20-s + (−46.5 − 80.6i)22-s + (80.3 + 139. i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.223 + 0.387i)5-s + (0.797 − 1.38i)7-s − 0.353·8-s + 0.316·10-s + (0.637 − 1.10i)11-s + (−0.982 − 1.70i)13-s + (−0.563 − 0.976i)14-s + (−0.125 + 0.216i)16-s + 0.0647·17-s + 1.05·19-s + (0.111 − 0.193i)20-s + (−0.450 − 0.781i)22-s + (0.728 + 1.26i)23-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $-0.939 + 0.342i$
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: \((2,\ 810,\ (\ :3/2),\ -0.939 + 0.342i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.372723320\)
\(L(\frac12)\) \(\approx\) \(2.372723320\)
\(L(\frac{5}{2})\) 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 \( 1 + (-14.7 + 25.5i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-23.2 + 40.3i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (46.0 + 79.7i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 4.53T + 4.91e3T^{2} \)
19 \( 1 - 87.5T + 6.85e3T^{2} \)
23 \( 1 + (-80.3 - 139. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (120. - 209. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-1.34 - 2.32i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 20.6T + 5.06e4T^{2} \)
41 \( 1 + (250. + 434. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (147. - 255. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-239. + 414. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 243.T + 1.48e5T^{2} \)
59 \( 1 + (191. + 332. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (66.3 - 114. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (291. + 504. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 566.T + 3.57e5T^{2} \)
73 \( 1 + 839.T + 3.89e5T^{2} \)
79 \( 1 + (225. - 391. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (150. - 260. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 739.T + 7.04e5T^{2} \)
97 \( 1 + (-573. + 992. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−9.801090150636540114994315547924, −8.711886443509593230080971079611, −7.59401324090433816199716432751, −7.07852514806480743918840527748, −5.58117719335816375192099480190, −5.07783319709279828313147782126, −3.64720371272644085188027419640, −3.14956430565201335115955143027, −1.49853014540636032822451673114, −0.56423899656516868160331927961, 1.67534947888865275266488548495, 2.57429306105165296194643682054, 4.38333240597545865890865331192, 4.80755881813316000049622510092, 5.78537318047053693641706761061, 6.76539190392560485003551720561, 7.54625025825495311890202727640, 8.630096183606603004165746772607, 9.242159626452621162603052786476, 9.845479793344946880381716239342

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