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Properties

Label 2-810-45.29-c2-0-41
Degree $2$
Conductor $810$
Sign $-0.422 + 0.906i$
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 + (1.29 − 4.82i)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 + (−9.65 + 2.58i)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 + (0.258 − 0.965i)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 + (−0.482 + 0.129i)20-s + (−0.0787 + 0.0454i)22-s + (0.0307 + 0.0532i)23-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8327004053\)
\(L(\frac12)\) \(\approx\) \(0.8327004053\)
\(L(2)\) 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 \( 1 + (-1.29 + 4.82i)T \)
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

−9.640626725672405460201460662300, −8.889842566501639989344214460129, −8.183340658933053376995026881801, −7.19152519149595183807301621188, −6.31098064786975394624369349968, −5.44127301194494257820755683966, −4.53787282000910270194154483894, −3.34880818431540144540785073698, −1.57788808176341063957022886781, −0.32299818180651178731216564562, 1.56289154390385078560417499282, 2.86001927329270523351376435066, 3.48646593671526344318068482886, 4.86086629676390212204181965860, 6.19576892789196389780338927266, 6.77054613552127540819399854408, 7.84834195809093025299226595423, 8.864909807997120253289941361785, 9.584560461684758583673364957069, 10.31041264913995990173825424166

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