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Properties

Label 2-810-9.7-c3-0-9
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 + (−7 + 12.1i)7-s − 7.99·8-s − 10·10-s + (−1.5 + 2.59i)11-s + (−23.5 − 40.7i)13-s + (14 + 24.2i)14-s + (−8 + 13.8i)16-s − 39·17-s + 32·19-s + (−10 + 17.3i)20-s + (3 + 5.19i)22-s + (49.5 + 85.7i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.223 − 0.387i)5-s + (−0.377 + 0.654i)7-s − 0.353·8-s − 0.316·10-s + (−0.0411 + 0.0712i)11-s + (−0.501 − 0.868i)13-s + (0.267 + 0.462i)14-s + (−0.125 + 0.216i)16-s − 0.556·17-s + 0.386·19-s + (−0.111 + 0.193i)20-s + (0.0290 + 0.0503i)22-s + (0.448 + 0.777i)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

\(L(2)\) \(\approx\) \(1.456005631\)
\(L(\frac12)\) \(\approx\) \(1.456005631\)
\(L(\frac{5}{2})\) 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 \( 1 + (7 - 12.1i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (23.5 + 40.7i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 39T + 4.91e3T^{2} \)
19 \( 1 - 32T + 6.85e3T^{2} \)
23 \( 1 + (-49.5 - 85.7i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (25.5 - 44.1i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (41.5 + 71.8i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 314T + 5.06e4T^{2} \)
41 \( 1 + (-54 - 93.5i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (149.5 - 258. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (265.5 - 459. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 564T + 1.48e5T^{2} \)
59 \( 1 + (6 + 10.3i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (115 - 199. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-134 - 232. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 120T + 3.57e5T^{2} \)
73 \( 1 - 1.10e3T + 3.89e5T^{2} \)
79 \( 1 + (-369.5 + 639. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (543 - 940. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 120T + 7.04e5T^{2} \)
97 \( 1 + (-821 + 1.42e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−9.764608830913919356627906031424, −9.345485164814553059047130600208, −8.300244597977675391919936349094, −7.40628084820660590996618090734, −6.15747063606377922517867111551, −5.35448569699092403208024011922, −4.48409181020971865935417330692, −3.29194227301197728120764433060, −2.42291469253189126017682938706, −0.986754177133864274022430160216, 0.40843114839121122645625447203, 2.30287016809113354940713719062, 3.55953139103198219434543786360, 4.36897766434579705916657929365, 5.36858874037411943107665746554, 6.65602663990705078774340325531, 6.93898204683007204984349142594, 7.926779977495317475216870007952, 8.886703673602922947762736687169, 9.753326134012964541545393152693

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