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Properties

Label 2-810-9.7-c3-0-17
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: $\chi_{810} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 810,\ (\ :3/2),\ -0.342 - 0.939i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.625198021\)
\(L(\frac12)\) \(\approx\) \(1.625198021\)
\(L(\frac{5}{2})\) 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

−9.790754608372828456717443392921, −9.431541777564165488719104500218, −8.262421144852842922665955943422, −7.57509060835275679542457253667, −6.61270544250175361015383787509, −6.06044576510155874253326509824, −4.72640984925324248752933458731, −3.96629575911547713518309880728, −2.35426446617006283791048799249, −1.12583231332011607045060665055, 0.56613371664451585518440972690, 1.65692237970406508114584582953, 2.98887069977566120199547499472, 3.80813956540309576160602551207, 5.43968358057916412859661275599, 5.62766307884982297430548031609, 7.29053059744198017467239158651, 8.151380898511842388781319180160, 8.740689008630671377366000959571, 9.676052442508948046541273537771

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