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Properties

Label 2-810-45.14-c2-0-9
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: $\chi_{810} (539, \cdot )$
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

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

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