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Properties

Label 2-270-45.4-c1-0-3
Degree $2$
Conductor $270$
Sign $0.358 + 0.933i$
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  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (−2.03 − 0.917i)5-s + (3.85 − 2.22i)7-s − 0.999i·8-s + (−2.22 + 0.224i)10-s + (0.724 + 1.25i)11-s + (−2.12 − 1.22i)13-s + (2.22 − 3.85i)14-s + (−0.5 − 0.866i)16-s − 3.89i·17-s + 0.550·19-s + (−1.81 + 1.30i)20-s + (1.25 + 0.724i)22-s + (2.51 + 1.44i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.911 − 0.410i)5-s + (1.45 − 0.840i)7-s − 0.353i·8-s + (−0.703 + 0.0710i)10-s + (0.218 + 0.378i)11-s + (−0.588 − 0.339i)13-s + (0.594 − 1.02i)14-s + (−0.125 − 0.216i)16-s − 0.945i·17-s + 0.126·19-s + (−0.405 + 0.292i)20-s + (0.267 + 0.154i)22-s + (0.523 + 0.302i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.358 + 0.933i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.358 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(270\)    =    \(2 \cdot 3^{3} \cdot 5\)
Sign: $0.358 + 0.933i$
Analytic conductor: \(2.15596\)
Root analytic conductor: \(1.46831\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{270} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 270,\ (\ :1/2),\ 0.358 + 0.933i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40520 - 0.965175i\)
\(L(\frac12)\) \(\approx\) \(1.40520 - 0.965175i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 \)
5 \( 1 + (2.03 + 0.917i)T \)
good7 \( 1 + (-3.85 + 2.22i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.724 - 1.25i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.12 + 1.22i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 3.89iT - 17T^{2} \)
19 \( 1 - 0.550T + 19T^{2} \)
23 \( 1 + (-2.51 - 1.44i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.22 - 5.58i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 8iT - 37T^{2} \)
41 \( 1 + (0.5 - 0.866i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.45 - 3.72i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.389 - 0.224i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 8.44iT - 53T^{2} \)
59 \( 1 + (-5.62 + 9.74i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.224 - 0.389i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.18 + 4.72i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.44T + 71T^{2} \)
73 \( 1 - 4.79iT - 73T^{2} \)
79 \( 1 + (3.67 + 6.36i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.46 - 2i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 12.8T + 89T^{2} \)
97 \( 1 + (-11.2 + 6.5i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.71179842642507928199538070847, −11.10583017631257555264278781896, −10.12703826638674050280604584361, −8.769380367516569302177933813527, −7.68737918511961265712076278036, −7.00138386962938504232079948450, −4.97493702114618585170040199619, −4.70252783571367168538656491759, −3.28478569587155588950003318853, −1.31719190468084963807392995878, 2.28986650408155220211641259389, 3.87553531927676286346684933667, 4.87375885390387584302169928952, 5.98500896218576177942925969311, 7.26375727050273025308314438383, 8.110599232399543540230151847933, 8.853960232478552011201389349233, 10.54286153534925670757789508239, 11.56142623897890784685765298204, 11.85061823908703780720148329234

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