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Properties

Label 2-810-15.8-c1-0-21
Degree $2$
Conductor $810$
Sign $-0.347 + 0.937i$
Analytic cond. $6.46788$
Root an. cond. $2.54320$
Motivic weight $1$
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 + 0.707i)2-s + 1.00i·4-s + (−1.86 − 1.23i)5-s + (−1.70 + 1.70i)7-s + (−0.707 + 0.707i)8-s + (−0.442 − 2.19i)10-s − 1.14i·11-s + (−1.74 − 1.74i)13-s − 2.40·14-s − 1.00·16-s + (−4.99 − 4.99i)17-s − 2.78i·19-s + (1.23 − 1.86i)20-s + (0.809 − 0.809i)22-s + (4.36 − 4.36i)23-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + 0.500i·4-s + (−0.833 − 0.553i)5-s + (−0.642 + 0.642i)7-s + (−0.250 + 0.250i)8-s + (−0.139 − 0.693i)10-s − 0.345i·11-s + (−0.485 − 0.485i)13-s − 0.642·14-s − 0.250·16-s + (−1.21 − 1.21i)17-s − 0.638i·19-s + (0.276 − 0.416i)20-s + (0.172 − 0.172i)22-s + (0.909 − 0.909i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.347 + 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.347 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $-0.347 + 0.937i$
Analytic conductor: \(6.46788\)
Root analytic conductor: \(2.54320\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{810} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 810,\ (\ :1/2),\ -0.347 + 0.937i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.283420 - 0.407066i\)
\(L(\frac12)\) \(\approx\) \(0.283420 - 0.407066i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 \)
5 \( 1 + (1.86 + 1.23i)T \)
good7 \( 1 + (1.70 - 1.70i)T - 7iT^{2} \)
11 \( 1 + 1.14iT - 11T^{2} \)
13 \( 1 + (1.74 + 1.74i)T + 13iT^{2} \)
17 \( 1 + (4.99 + 4.99i)T + 17iT^{2} \)
19 \( 1 + 2.78iT - 19T^{2} \)
23 \( 1 + (-4.36 + 4.36i)T - 23iT^{2} \)
29 \( 1 + 1.34T + 29T^{2} \)
31 \( 1 + 2.50T + 31T^{2} \)
37 \( 1 + (8.16 - 8.16i)T - 37iT^{2} \)
41 \( 1 + 1.97iT - 41T^{2} \)
43 \( 1 + (6.35 + 6.35i)T + 43iT^{2} \)
47 \( 1 + (8.72 + 8.72i)T + 47iT^{2} \)
53 \( 1 + (-1.84 + 1.84i)T - 53iT^{2} \)
59 \( 1 + 2.62T + 59T^{2} \)
61 \( 1 - 7.08T + 61T^{2} \)
67 \( 1 + (0.0399 - 0.0399i)T - 67iT^{2} \)
71 \( 1 - 9.10iT - 71T^{2} \)
73 \( 1 + (-7.82 - 7.82i)T + 73iT^{2} \)
79 \( 1 - 9.77iT - 79T^{2} \)
83 \( 1 + (-1.98 + 1.98i)T - 83iT^{2} \)
89 \( 1 + 4.87T + 89T^{2} \)
97 \( 1 + (-5.70 + 5.70i)T - 97iT^{2} \)
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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

−9.827329742137320520484555992219, −8.786568113465134548292843729176, −8.477322468483695043339990928871, −7.12762615242570493469813671135, −6.71137242308416593885391668467, −5.28476679735240667140669180217, −4.83137034515960455262467313154, −3.54974824791454236607439300444, −2.60898329377865589798773675898, −0.19696937605621904112552823985, 1.86814080935002729168001847055, 3.30093884028834231012414015900, 3.94043499641000083608847629606, 4.88270827079224436073461357652, 6.28571170681968801677680153205, 6.94643417231554815402536761155, 7.78718722523055706442235126376, 8.975989236698145076715490314097, 9.873749413224757105738041446861, 10.71578376580636163957864428215

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