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Properties

Label 2-810-45.29-c2-0-5
Degree $2$
Conductor $810$
Sign $0.207 - 0.978i$
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 + (−4.24 − 2.64i)5-s + (−11.8 − 6.84i)7-s + 2.82·8-s + (6.24 − 3.32i)10-s + (−10.6 − 6.15i)11-s + (−14.7 + 8.50i)13-s + (16.7 − 9.67i)14-s + (−2.00 + 3.46i)16-s + 6.89·17-s − 7.24·19-s + (−0.346 + 9.99i)20-s + (15.0 − 8.70i)22-s + (−17.3 − 30.1i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.848 − 0.529i)5-s + (−1.69 − 0.977i)7-s + 0.353·8-s + (0.624 − 0.332i)10-s + (−0.969 − 0.559i)11-s + (−1.13 + 0.654i)13-s + (1.19 − 0.691i)14-s + (−0.125 + 0.216i)16-s + 0.405·17-s − 0.381·19-s + (−0.0173 + 0.499i)20-s + (0.685 − 0.395i)22-s + (−0.756 − 1.31i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.207 - 0.978i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.207 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $0.207 - 0.978i$
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: \((2,\ 810,\ (\ :1),\ 0.207 - 0.978i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2461122161\)
\(L(\frac12)\) \(\approx\) \(0.2461122161\)
\(L(2)\) 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 \( 1 + (4.24 + 2.64i)T \)
good7 \( 1 + (11.8 + 6.84i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (10.6 + 6.15i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (14.7 - 8.50i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 - 6.89T + 289T^{2} \)
19 \( 1 + 7.24T + 361T^{2} \)
23 \( 1 + (17.3 + 30.1i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-18.3 - 10.5i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-19.1 - 33.0i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 21.5iT - 1.36e3T^{2} \)
41 \( 1 + (-31.4 + 18.1i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (5.40 + 3.11i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-20.1 + 34.8i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 38.2T + 2.80e3T^{2} \)
59 \( 1 + (36.0 - 20.8i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-7.52 + 13.0i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (111. - 64.3i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 104. iT - 5.04e3T^{2} \)
73 \( 1 - 2.11iT - 5.32e3T^{2} \)
79 \( 1 + (-22.0 + 38.1i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-27.5 + 47.6i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 68.1iT - 7.92e3T^{2} \)
97 \( 1 + (87.6 + 50.6i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−10.28956798494027366298111407299, −9.286196382897542205465696607985, −8.498341823528658141168720608510, −7.53999544229502545546858311238, −6.96085629642522510578021725670, −6.06630321402146009790274767316, −4.83245970543263564561203747742, −3.96991360697020507313288918671, −2.81393725788715210346190998928, −0.59573891358505706121519967929, 0.17087934155385081822813630987, 2.55523594474710831199619177002, 2.94399051906611577100944918549, 4.12334414334476920317549091466, 5.43851261271415473900831965462, 6.45275355638705832968510965928, 7.54092451314877124607053673261, 8.060892410297131555419314410956, 9.331659504257992193053940161893, 9.933369447760236578170390557726

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