Properties

Label 2-810-45.14-c2-0-5
Degree $2$
Conductor $810$
Sign $0.413 - 0.910i$
Analytic cond. $22.0709$
Root an. cond. $4.69796$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 − 1.22i)2-s + (−0.999 + 1.73i)4-s + (−2.15 − 4.51i)5-s + (5.04 − 2.91i)7-s + 2.82·8-s + (−4 + 5.83i)10-s + (−14.2 + 8.24i)11-s + (−7.14 − 4.12i)14-s + (−2.00 − 3.46i)16-s − 11.3·17-s + 12·19-s + (9.96 + 0.775i)20-s + (20.1 + 11.6i)22-s + (−12.0 + 20.8i)23-s + (−15.6 + 19.4i)25-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.431 − 0.902i)5-s + (0.721 − 0.416i)7-s + 0.353·8-s + (−0.400 + 0.583i)10-s + (−1.29 + 0.749i)11-s + (−0.510 − 0.294i)14-s + (−0.125 − 0.216i)16-s − 0.665·17-s + 0.631·19-s + (0.498 + 0.0387i)20-s + (0.918 + 0.530i)22-s + (−0.522 + 0.905i)23-s + (−0.627 + 0.778i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.413 - 0.910i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.413 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $0.413 - 0.910i$
Analytic conductor: \(22.0709\)
Root analytic conductor: \(4.69796\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{810} (539, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 810,\ (\ :1),\ 0.413 - 0.910i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5065692414\)
\(L(\frac12)\) \(\approx\) \(0.5065692414\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 1.22i)T \)
3 \( 1 \)
5 \( 1 + (2.15 + 4.51i)T \)
good7 \( 1 + (-5.04 + 2.91i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (14.2 - 8.24i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (84.5 + 146. i)T^{2} \)
17 \( 1 + 11.3T + 289T^{2} \)
19 \( 1 - 12T + 361T^{2} \)
23 \( 1 + (12.0 - 20.8i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-16 + 27.7i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 23.3iT - 1.36e3T^{2} \)
41 \( 1 + (49.9 + 28.8i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (35.3 - 20.4i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-17.6 - 30.6i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 67.8T + 2.80e3T^{2} \)
59 \( 1 + (-14.2 - 8.24i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-8 - 13.8i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-5.04 - 2.91i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 - 116. iT - 5.32e3T^{2} \)
79 \( 1 + (-36 - 62.3i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-21.9 - 37.9i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 65.9iT - 7.92e3T^{2} \)
97 \( 1 + (141. - 81.6i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−10.13663544080566960379067902335, −9.510308666054898605708889640501, −8.389502194978815356081043915521, −7.908718948147017619138935871442, −7.12326572511735349812000158373, −5.43375898823392580501123366094, −4.73088964002327242399674216415, −3.85133355127421471648040228285, −2.39361733205909810586739799038, −1.22476300992649258849215577222, 0.20487223769720513872163339257, 2.17169935596389801538577890400, 3.29911784583145984509211723343, 4.70342096140153723240161635152, 5.55367799495795713612726075484, 6.53531107107094703437885391266, 7.38177114041431807554144959745, 8.246773893891077187760081728809, 8.652607676700973553795781441689, 10.06372774615341020494590356557

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