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Properties

Label 2-270-45.32-c1-0-0
Degree $2$
Conductor $270$
Sign $0.947 - 0.320i$
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  + (−0.965 − 0.258i)2-s + (0.866 + 0.499i)4-s + (2.22 + 0.210i)5-s + (−0.521 + 1.94i)7-s + (−0.707 − 0.707i)8-s + (−2.09 − 0.779i)10-s + (1.70 − 0.984i)11-s + (1.05 + 3.92i)13-s + (1.00 − 1.74i)14-s + (0.500 + 0.866i)16-s + (−2.35 + 2.35i)17-s − 3.70i·19-s + (1.82 + 1.29i)20-s + (−1.90 + 0.509i)22-s + (6.05 − 1.62i)23-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (0.433 + 0.249i)4-s + (0.995 + 0.0942i)5-s + (−0.197 + 0.736i)7-s + (−0.249 − 0.249i)8-s + (−0.662 − 0.246i)10-s + (0.514 − 0.296i)11-s + (0.291 + 1.08i)13-s + (0.269 − 0.466i)14-s + (0.125 + 0.216i)16-s + (−0.572 + 0.572i)17-s − 0.850i·19-s + (0.407 + 0.289i)20-s + (−0.405 + 0.108i)22-s + (1.26 − 0.338i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.320i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 - 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(270\)    =    \(2 \cdot 3^{3} \cdot 5\)
Sign: $0.947 - 0.320i$
Analytic conductor: \(2.15596\)
Root analytic conductor: \(1.46831\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{270} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 270,\ (\ :1/2),\ 0.947 - 0.320i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08518 + 0.178464i\)
\(L(\frac12)\) \(\approx\) \(1.08518 + 0.178464i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 + 0.258i)T \)
3 \( 1 \)
5 \( 1 + (-2.22 - 0.210i)T \)
good7 \( 1 + (0.521 - 1.94i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-1.70 + 0.984i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.05 - 3.92i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (2.35 - 2.35i)T - 17iT^{2} \)
19 \( 1 + 3.70iT - 19T^{2} \)
23 \( 1 + (-6.05 + 1.62i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-3.74 - 6.49i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.48 + 6.04i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.26 - 4.26i)T + 37iT^{2} \)
41 \( 1 + (6.13 + 3.54i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (9.09 + 2.43i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (7.49 + 2.00i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (7.03 + 7.03i)T + 53iT^{2} \)
59 \( 1 + (1.34 - 2.33i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.37 + 7.57i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.18 - 2.19i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 5.68iT - 71T^{2} \)
73 \( 1 + (-1.14 + 1.14i)T - 73iT^{2} \)
79 \( 1 + (10.0 - 5.80i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.440 + 1.64i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 - 2.04T + 89T^{2} \)
97 \( 1 + (-2.60 + 9.71i)T + (-84.0 - 48.5i)T^{2} \)
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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

−11.73801313859923238680515031902, −11.00063844328143468263843479217, −9.927135668651686983890294694682, −9.029372426522131791814841898923, −8.599284937149355688416063059351, −6.79677424974318469856111458326, −6.31332194353673248446704910602, −4.84824754653915857411986905295, −3.00989689381633690000817030054, −1.70886409448940139643272507284, 1.26734472043361472196896565622, 3.01516917101864514759624363186, 4.81119594454174302354704026470, 6.08846356225851822554698250066, 6.91793524147390221032750028655, 8.061079049520040115666364272470, 9.117642975261623487483674065999, 9.998824653706070292427331709709, 10.56584774939150583902742207131, 11.71132532350012632193818178862

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