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Properties

Label 2-810-45.14-c2-0-35
Degree $2$
Conductor $810$
Sign $0.906 - 0.422i$
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 + (4.82 − 1.29i)5-s + (4.33 − 2.5i)7-s − 2.82·8-s + (5 + 5i)10-s + (1.22 − 0.707i)11-s + (−7.79 − 4.5i)13-s + (6.12 + 3.53i)14-s + (−2.00 − 3.46i)16-s + 11.3·17-s + 21·19-s + (−2.58 + 9.65i)20-s + (1.73 + 0.999i)22-s + (−0.707 + 1.22i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.965 − 0.258i)5-s + (0.618 − 0.357i)7-s − 0.353·8-s + (0.5 + 0.5i)10-s + (0.111 − 0.0642i)11-s + (−0.599 − 0.346i)13-s + (0.437 + 0.252i)14-s + (−0.125 − 0.216i)16-s + 0.665·17-s + 1.10·19-s + (−0.129 + 0.482i)20-s + (0.0787 + 0.0454i)22-s + (−0.0307 + 0.0532i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 - 0.422i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.906 - 0.422i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $0.906 - 0.422i$
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: \((2,\ 810,\ (\ :1),\ 0.906 - 0.422i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.890520856\)
\(L(\frac12)\) \(\approx\) \(2.890520856\)
\(L(2)\) 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 \( 1 + (-4.82 + 1.29i)T \)
good7 \( 1 + (-4.33 + 2.5i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (-1.22 + 0.707i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (7.79 + 4.5i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 - 11.3T + 289T^{2} \)
19 \( 1 - 21T + 361T^{2} \)
23 \( 1 + (0.707 - 1.22i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-33.0 + 19.0i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (20 - 34.6i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 25iT - 1.36e3T^{2} \)
41 \( 1 + (45.3 + 26.1i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-55.4 + 32i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (11.3 + 19.5i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 72.1T + 2.80e3T^{2} \)
59 \( 1 + (78.3 + 45.2i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-48.5 - 84.0i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-113. - 65.5i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 89.0iT - 5.04e3T^{2} \)
73 \( 1 + 17iT - 5.32e3T^{2} \)
79 \( 1 + (-58.5 - 101. i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-28.9 - 50.2i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 147. iT - 7.92e3T^{2} \)
97 \( 1 + (35.5 - 20.5i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−10.02414018875986900083713525376, −9.225384797332070877195699473128, −8.302832178342195139252228305355, −7.45431736544125754388773096326, −6.63600861702145078422940766846, −5.43703337284921122529567510454, −5.12855301265587452773251623240, −3.83188437477666988648593818264, −2.52691125427826533948238085525, −1.06215654488537593490921724763, 1.25410774572207089497023373181, 2.30366631539596373474114719061, 3.28643877428678392925611519603, 4.72142992505554839287086640206, 5.38043784931938632047793799686, 6.30356563386831585182800362347, 7.35889421736757348226001852943, 8.450007827049621018085959884306, 9.513904569817348115276025942288, 9.895536750161165185433877981082

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