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Properties

Label 2-270-5.4-c1-0-5
Degree $2$
Conductor $270$
Sign $0.974 - 0.223i$
Analytic cond. $2.15596$
Root an. cond. $1.46831$
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 + (2.17 − 0.5i)5-s − 4.35i·7-s − i·8-s + (0.5 + 2.17i)10-s + 4.35·11-s + 4.35·14-s + 16-s + 4i·17-s − 6·19-s + (−2.17 + 0.5i)20-s + 4.35i·22-s + 2i·23-s + (4.50 − 2.17i)25-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (0.974 − 0.223i)5-s − 1.64i·7-s − 0.353i·8-s + (0.158 + 0.689i)10-s + 1.31·11-s + 1.16·14-s + 0.250·16-s + 0.970i·17-s − 1.37·19-s + (−0.487 + 0.111i)20-s + 0.929i·22-s + 0.417i·23-s + (0.900 − 0.435i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.223i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 - 0.223i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(270\)    =    \(2 \cdot 3^{3} \cdot 5\)
Sign: $0.974 - 0.223i$
Analytic conductor: \(2.15596\)
Root analytic conductor: \(1.46831\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{270} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 270,\ (\ :1/2),\ 0.974 - 0.223i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42295 + 0.161130i\)
\(L(\frac12)\) \(\approx\) \(1.42295 + 0.161130i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.17 + 0.5i)T \)
good7 \( 1 + 4.35iT - 7T^{2} \)
11 \( 1 - 4.35T + 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 4iT - 17T^{2} \)
19 \( 1 + 6T + 19T^{2} \)
23 \( 1 - 2iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 7T + 31T^{2} \)
37 \( 1 - 8.71iT - 37T^{2} \)
41 \( 1 + 8.71T + 41T^{2} \)
43 \( 1 + 8.71iT - 43T^{2} \)
47 \( 1 - 2iT - 47T^{2} \)
53 \( 1 - 3iT - 53T^{2} \)
59 \( 1 + 8.71T + 59T^{2} \)
61 \( 1 + 4T + 61T^{2} \)
67 \( 1 + 8.71iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 4.35iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 5iT - 83T^{2} \)
89 \( 1 - 8.71T + 89T^{2} \)
97 \( 1 - 4.35iT - 97T^{2} \)
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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

−12.11978981791208913163756420069, −10.66332091886511979844005050481, −10.07713811312692383146876162396, −9.020531779319398921644257226420, −8.053229304902307451834733415791, −6.70548239590063049571171499498, −6.31807200379814711901959686957, −4.73120116441461174560227768954, −3.79686094233131321407077484596, −1.39479614281458716766363622217, 1.89475710595397661077940748203, 2.90956297276339527698007690151, 4.63088320330289034880030398323, 5.83624473813148852273036815983, 6.63369949335041670626880174907, 8.525631392692962727035906310322, 9.146929222652268802095458709429, 9.879451865879765104275725031949, 11.05841395546642375152757259305, 11.94015517362948573116104937555

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