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Properties

Label 2-810-45.29-c2-0-30
Degree $2$
Conductor $810$
Sign $0.711 + 0.702i$
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 + (4.91 − 0.897i)5-s + (8.15 + 4.70i)7-s − 2.82·8-s + (2.37 − 6.65i)10-s + (−2.03 − 1.17i)11-s + (1.33 − 0.769i)13-s + (11.5 − 6.65i)14-s + (−2.00 + 3.46i)16-s + 11.0·17-s + 7.09·19-s + (−6.47 − 7.62i)20-s + (−2.88 + 1.66i)22-s + (4.09 + 7.09i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.983 − 0.179i)5-s + (1.16 + 0.672i)7-s − 0.353·8-s + (0.237 − 0.665i)10-s + (−0.185 − 0.107i)11-s + (0.102 − 0.0591i)13-s + (0.823 − 0.475i)14-s + (−0.125 + 0.216i)16-s + 0.652·17-s + 0.373·19-s + (−0.323 − 0.381i)20-s + (−0.131 + 0.0756i)22-s + (0.178 + 0.308i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $0.711 + 0.702i$
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: \((2,\ 810,\ (\ :1),\ 0.711 + 0.702i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.094101597\)
\(L(\frac12)\) \(\approx\) \(3.094101597\)
\(L(2)\) 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 \( 1 + (-4.91 + 0.897i)T \)
good7 \( 1 + (-8.15 - 4.70i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (2.03 + 1.17i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-1.33 + 0.769i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 - 11.0T + 289T^{2} \)
19 \( 1 - 7.09T + 361T^{2} \)
23 \( 1 + (-4.09 - 7.09i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (15.5 + 8.97i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-29.3 - 50.8i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 20.7iT - 1.36e3T^{2} \)
41 \( 1 + (42.3 - 24.4i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-3.07 - 1.77i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-34.8 + 60.4i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 69.0T + 2.80e3T^{2} \)
59 \( 1 + (-34.0 + 19.6i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (57.8 - 100. i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (88.8 - 51.2i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 102. iT - 5.04e3T^{2} \)
73 \( 1 + 120. iT - 5.32e3T^{2} \)
79 \( 1 + (-9.13 + 15.8i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-80.9 + 140. i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 88.2iT - 7.92e3T^{2} \)
97 \( 1 + (-121. - 70.3i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−10.18511494390807800228941564206, −9.082576170720243850919288908194, −8.548816322342677602509135738273, −7.40924294714126305366413005040, −6.08883368265132961549336408220, −5.34167618609629320493918152644, −4.74203954724871564669703231007, −3.25290095588954109074246285497, −2.14476369701025592726024187589, −1.23412899315543432907826015553, 1.21278730290004027474941112302, 2.55602184037199283956654604507, 3.96654418610464931957877063958, 4.96558453053592532058170452527, 5.66944572900025849960120616225, 6.66286967229640580052873837902, 7.55612873036473570165806560067, 8.234155830542235207355533383573, 9.293087529172007462199123219881, 10.13248715721769480105973967219

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