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Properties

Label 2-810-45.14-c2-0-46
Degree $2$
Conductor $810$
Sign $-0.759 + 0.650i$
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 + (1.24 − 4.84i)5-s + (0.315 − 0.182i)7-s − 2.82·8-s + (6.81 − 1.89i)10-s + (−8.32 + 4.80i)11-s + (−6.62 − 3.82i)13-s + (0.446 + 0.257i)14-s + (−2.00 − 3.46i)16-s − 12.3·17-s + 5.36·19-s + (7.13 + 7.00i)20-s + (−11.7 − 6.79i)22-s + (6.03 − 10.4i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.249 − 0.968i)5-s + (0.0451 − 0.0260i)7-s − 0.353·8-s + (0.681 − 0.189i)10-s + (−0.756 + 0.436i)11-s + (−0.509 − 0.294i)13-s + (0.0319 + 0.0184i)14-s + (−0.125 − 0.216i)16-s − 0.725·17-s + 0.282·19-s + (0.356 + 0.350i)20-s + (−0.535 − 0.308i)22-s + (0.262 − 0.454i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.759 + 0.650i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.759 + 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $-0.759 + 0.650i$
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: \((2,\ 810,\ (\ :1),\ -0.759 + 0.650i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2573611937\)
\(L(\frac12)\) \(\approx\) \(0.2573611937\)
\(L(2)\) 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 \( 1 + (-1.24 + 4.84i)T \)
good7 \( 1 + (-0.315 + 0.182i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (8.32 - 4.80i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (6.62 + 3.82i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + 12.3T + 289T^{2} \)
19 \( 1 - 5.36T + 361T^{2} \)
23 \( 1 + (-6.03 + 10.4i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (27.1 - 15.6i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (2.45 - 4.26i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 40.2iT - 1.36e3T^{2} \)
41 \( 1 + (54.8 + 31.6i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (46.6 - 26.9i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (14.1 + 24.4i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 41.6T + 2.80e3T^{2} \)
59 \( 1 + (97.2 + 56.1i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (22.9 + 39.8i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-39.9 - 23.0i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 125. iT - 5.04e3T^{2} \)
73 \( 1 - 59.0iT - 5.32e3T^{2} \)
79 \( 1 + (25.3 + 43.8i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (29.5 + 51.1i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 8.93iT - 7.92e3T^{2} \)
97 \( 1 + (-113. + 65.6i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−9.588217102715040543787982701666, −8.740788809270681097941880815824, −7.998614822607280519811962543784, −7.14205905801312940371033716137, −6.13519614488827078928777078194, −5.02380840519107434914606232909, −4.74236535798068888284551755538, −3.29080179910885283204650351323, −1.87766905659915712880172557511, −0.07006919435104475190732531422, 1.90276773977589175455866059249, 2.83565832523334428254524830299, 3.78577756374530005274729165196, 5.02463370157867728968221343296, 5.87971417537444865495552527445, 6.87318005364921460379191622410, 7.72072845502238504071917656429, 8.881448002759352649875799855196, 9.797540998020982289992905858836, 10.43475003917346385283327995373

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