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Properties

Label 2-810-9.7-c3-0-27
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 + (−4 + 6.92i)7-s + 7.99·8-s + 10·10-s + (−9 + 15.5i)11-s + (−4 − 6.92i)13-s + (−7.99 − 13.8i)14-s + (−8 + 13.8i)16-s + 15·17-s + 23·19-s + (−10 + 17.3i)20-s + (−18 − 31.1i)22-s + (−31.5 − 54.5i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.223 − 0.387i)5-s + (−0.215 + 0.374i)7-s + 0.353·8-s + 0.316·10-s + (−0.246 + 0.427i)11-s + (−0.0853 − 0.147i)13-s + (−0.152 − 0.264i)14-s + (−0.125 + 0.216i)16-s + 0.214·17-s + 0.277·19-s + (−0.111 + 0.193i)20-s + (−0.174 − 0.302i)22-s + (−0.285 − 0.494i)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

\(L(2)\) \(\approx\) \(0.9747707201\)
\(L(\frac12)\) \(\approx\) \(0.9747707201\)
\(L(\frac{5}{2})\) 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 \( 1 + (4 - 6.92i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (9 - 15.5i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (4 + 6.92i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 15T + 4.91e3T^{2} \)
19 \( 1 - 23T + 6.85e3T^{2} \)
23 \( 1 + (31.5 + 54.5i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (78 - 135. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-42.5 - 73.6i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 74T + 5.06e4T^{2} \)
41 \( 1 + (123 + 213. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-95 + 164. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (144 - 249. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 177T + 1.48e5T^{2} \)
59 \( 1 + (396 + 685. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-453.5 + 785. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-161 - 278. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 270T + 3.57e5T^{2} \)
73 \( 1 - 254T + 3.89e5T^{2} \)
79 \( 1 + (-561.5 + 972. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-385.5 + 667. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 198T + 7.04e5T^{2} \)
97 \( 1 + (-596 + 1.03e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−9.589263134500389049187791585337, −8.877653529811197601754735928561, −8.041439148865973846017352169153, −7.28377423085863834280228840558, −6.32801607276344107310263535061, −5.37797893211592572723722348148, −4.58683054455506382053785843975, −3.28318997005936655546251710748, −1.83482592663656322006272390353, −0.37135654806475222985477573121, 0.917991322665324466677963375347, 2.34962687377700102969995727981, 3.38709069969464704809835919348, 4.24208159897327380499755455108, 5.51628985357374930417405942182, 6.59546514671959402346121703261, 7.58869955222606808943481712710, 8.213028539892875667248101309692, 9.327791720666643929281496050924, 9.988800965029397070343448580952

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