Properties

Label 2-810-15.14-c2-0-25
Degree $2$
Conductor $810$
Sign $0.999 + 0.00295i$
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  − 1.41·2-s + 2.00·4-s + (4.99 + 0.0147i)5-s + 5.91i·7-s − 2.82·8-s + (−7.07 − 0.0209i)10-s + 3.80i·11-s − 22.2i·13-s − 8.37i·14-s + 4.00·16-s − 1.20·17-s + 29.3·19-s + (9.99 + 0.0295i)20-s − 5.38i·22-s − 16.1·23-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.500·4-s + (0.999 + 0.00295i)5-s + 0.845i·7-s − 0.353·8-s + (−0.707 − 0.00209i)10-s + 0.346i·11-s − 1.70i·13-s − 0.597i·14-s + 0.250·16-s − 0.0708·17-s + 1.54·19-s + (0.499 + 0.00147i)20-s − 0.244i·22-s − 0.701·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00295i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.00295i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $0.999 + 0.00295i$
Analytic conductor: \(22.0709\)
Root analytic conductor: \(4.69796\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{810} (809, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 810,\ (\ :1),\ 0.999 + 0.00295i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.683426961\)
\(L(\frac12)\) \(\approx\) \(1.683426961\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41T \)
3 \( 1 \)
5 \( 1 + (-4.99 - 0.0147i)T \)
good7 \( 1 - 5.91iT - 49T^{2} \)
11 \( 1 - 3.80iT - 121T^{2} \)
13 \( 1 + 22.2iT - 169T^{2} \)
17 \( 1 + 1.20T + 289T^{2} \)
19 \( 1 - 29.3T + 361T^{2} \)
23 \( 1 + 16.1T + 529T^{2} \)
29 \( 1 + 45.2iT - 841T^{2} \)
31 \( 1 - 43.7T + 961T^{2} \)
37 \( 1 - 48.4iT - 1.36e3T^{2} \)
41 \( 1 + 2.72iT - 1.68e3T^{2} \)
43 \( 1 - 19.9iT - 1.84e3T^{2} \)
47 \( 1 + 24.2T + 2.20e3T^{2} \)
53 \( 1 + 10.8T + 2.80e3T^{2} \)
59 \( 1 - 19.3iT - 3.48e3T^{2} \)
61 \( 1 - 18.3T + 3.72e3T^{2} \)
67 \( 1 + 83.1iT - 4.48e3T^{2} \)
71 \( 1 - 121. iT - 5.04e3T^{2} \)
73 \( 1 + 105. iT - 5.32e3T^{2} \)
79 \( 1 - 94.2T + 6.24e3T^{2} \)
83 \( 1 - 94.6T + 6.88e3T^{2} \)
89 \( 1 - 132. iT - 7.92e3T^{2} \)
97 \( 1 - 119. iT - 9.40e3T^{2} \)
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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

−9.864233700841012595639180759896, −9.462399863825466234011351739252, −8.294272872650435280844994931161, −7.77734136722072186336218047822, −6.46602929134701859473659357083, −5.77283764738460332016242037451, −4.97910704833368313895445260400, −3.11937880149611837768507817219, −2.31581197621584133844560065183, −0.940374532615153761942675661547, 1.02448329330649477667799943123, 2.06125778886661761199035611486, 3.42695441041899184758757281797, 4.70404356677814480297188391873, 5.84072060070861677369714573205, 6.77470280558553006312280618880, 7.34935972327070404718574800009, 8.560693147810053422803586814667, 9.321262840357779635152542751661, 9.914058150661941690968751163745

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