Properties

Label 2-810-45.29-c2-0-25
Degree $2$
Conductor $810$
Sign $0.419 - 0.907i$
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 + (2.51 + 4.32i)5-s + (5.12 + 2.95i)7-s + 2.82·8-s + (−7.07 + 0.0209i)10-s + (−3.29 − 1.90i)11-s + (19.2 − 11.1i)13-s + (−7.24 + 4.18i)14-s + (−2.00 + 3.46i)16-s + 1.20·17-s + 29.3·19-s + (4.97 − 8.67i)20-s + (4.66 − 2.69i)22-s + (−8.07 − 13.9i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.502 + 0.864i)5-s + (0.732 + 0.422i)7-s + 0.353·8-s + (−0.707 + 0.00209i)10-s + (−0.299 − 0.173i)11-s + (1.48 − 0.854i)13-s + (−0.517 + 0.298i)14-s + (−0.125 + 0.216i)16-s + 0.0708·17-s + 1.54·19-s + (0.248 − 0.433i)20-s + (0.212 − 0.122i)22-s + (−0.350 − 0.607i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.419 - 0.907i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.419 - 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $0.419 - 0.907i$
Analytic conductor: \(22.0709\)
Root analytic conductor: \(4.69796\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{810} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 810,\ (\ :1),\ 0.419 - 0.907i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.004153452\)
\(L(\frac12)\) \(\approx\) \(2.004153452\)
\(L(2)\) 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 \( 1 + (-2.51 - 4.32i)T \)
good7 \( 1 + (-5.12 - 2.95i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (3.29 + 1.90i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-19.2 + 11.1i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 - 1.20T + 289T^{2} \)
19 \( 1 - 29.3T + 361T^{2} \)
23 \( 1 + (8.07 + 13.9i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-39.2 - 22.6i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (21.8 + 37.9i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 48.4iT - 1.36e3T^{2} \)
41 \( 1 + (2.35 - 1.36i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-17.3 - 9.99i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (12.1 - 21.0i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 10.8T + 2.80e3T^{2} \)
59 \( 1 + (-16.7 + 9.69i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (9.19 - 15.9i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-71.9 + 41.5i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 121. iT - 5.04e3T^{2} \)
73 \( 1 - 105. iT - 5.32e3T^{2} \)
79 \( 1 + (47.1 - 81.5i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-47.3 + 81.9i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 132. iT - 7.92e3T^{2} \)
97 \( 1 + (-103. - 59.5i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−10.19125370271163592556770160208, −9.292686764944788766106284315088, −8.354510876398342987996687387412, −7.73635932255122215619467774207, −6.73565456138536230501466251114, −5.78908176207091798433396097658, −5.29907607264321970479654759249, −3.73490412521746520892448087762, −2.54820273266303509281673427037, −1.10627216659610562004462746410, 1.04600337402953229991885389696, 1.75108214040205425280748482218, 3.33818489420361967686348503115, 4.44315069144517455560970816227, 5.24131980939195608387284974050, 6.38221288706934735793754057271, 7.61423092839809224077862577276, 8.386276940126388127836967860519, 9.081705087502570242890725926225, 9.900346011539389053540101699152

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