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Properties

Label 2-810-5.4-c1-0-2
Degree $2$
Conductor $810$
Sign $-0.994 + 0.100i$
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  + i·2-s − 4-s + (0.224 + 2.22i)5-s + 4.44i·7-s − i·8-s + (−2.22 + 0.224i)10-s − 1.44·11-s + 2.44i·13-s − 4.44·14-s + 16-s − 3.89i·17-s + 0.550·19-s + (−0.224 − 2.22i)20-s − 1.44i·22-s − 2.89i·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (0.100 + 0.994i)5-s + 1.68i·7-s − 0.353i·8-s + (−0.703 + 0.0710i)10-s − 0.437·11-s + 0.679i·13-s − 1.18·14-s + 0.250·16-s − 0.945i·17-s + 0.126·19-s + (−0.0502 − 0.497i)20-s − 0.309i·22-s − 0.604i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.100i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $-0.994 + 0.100i$
Analytic conductor: \(6.46788\)
Root analytic conductor: \(2.54320\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{810} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 810,\ (\ :1/2),\ -0.994 + 0.100i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0557667 - 1.10687i\)
\(L(\frac12)\) \(\approx\) \(0.0557667 - 1.10687i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
5 \( 1 + (-0.224 - 2.22i)T \)
good7 \( 1 - 4.44iT - 7T^{2} \)
11 \( 1 + 1.44T + 11T^{2} \)
13 \( 1 - 2.44iT - 13T^{2} \)
17 \( 1 + 3.89iT - 17T^{2} \)
19 \( 1 - 0.550T + 19T^{2} \)
23 \( 1 + 2.89iT - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 - 6.44T + 31T^{2} \)
37 \( 1 - 8iT - 37T^{2} \)
41 \( 1 - T + 41T^{2} \)
43 \( 1 + 7.44iT - 43T^{2} \)
47 \( 1 + 0.449iT - 47T^{2} \)
53 \( 1 - 8.44iT - 53T^{2} \)
59 \( 1 + 11.2T + 59T^{2} \)
61 \( 1 + 0.449T + 61T^{2} \)
67 \( 1 - 9.44iT - 67T^{2} \)
71 \( 1 + 2.44T + 71T^{2} \)
73 \( 1 - 4.79iT - 73T^{2} \)
79 \( 1 - 7.34T + 79T^{2} \)
83 \( 1 + 4iT - 83T^{2} \)
89 \( 1 - 12.8T + 89T^{2} \)
97 \( 1 - 13iT - 97T^{2} \)
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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

−10.59079819293682012575303393015, −9.612465379620211254762990013577, −8.992025176635735086833969454189, −8.086998021904927528567247646175, −7.15022043591505287788729675187, −6.31383930821302210802214724762, −5.61346477063421306841344834501, −4.64351137362728007332496379214, −3.12246169117477411058592468568, −2.24784914173674698392708835056, 0.55157875335657993810086183926, 1.69949810537922217096501280842, 3.40269581418943620161083475595, 4.21986075562604119682947416870, 5.08126892479478217535828235106, 6.16207523753907946407831573676, 7.61136366283093961010516600972, 8.012231247894455954764785455691, 9.169697959697546966513588545366, 9.962631419320365579046890799843

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