Properties

Label 2-270-9.5-c10-0-5
Degree $2$
Conductor $270$
Sign $0.646 - 0.763i$
Analytic cond. $171.546$
Root an. cond. $13.0975$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (19.5 − 11.3i)2-s + (255. − 443. i)4-s + (−1.21e3 − 698. i)5-s + (−5.11e3 − 8.85e3i)7-s − 1.15e4i·8-s − 3.16e4·10-s + (8.38e4 − 4.84e4i)11-s + (−5.74e4 + 9.95e4i)13-s + (−2.00e5 − 1.15e5i)14-s + (−1.31e5 − 2.27e5i)16-s + 1.26e6i·17-s + 1.73e6·19-s + (−6.19e5 + 3.57e5i)20-s + (1.09e6 − 1.89e6i)22-s + (−6.63e6 − 3.83e6i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.387 − 0.223i)5-s + (−0.304 − 0.526i)7-s − 0.353i·8-s − 0.316·10-s + (0.520 − 0.300i)11-s + (−0.154 + 0.268i)13-s + (−0.372 − 0.215i)14-s + (−0.125 − 0.216i)16-s + 0.893i·17-s + 0.701·19-s + (−0.193 + 0.111i)20-s + (0.212 − 0.368i)22-s + (−1.03 − 0.595i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.646 - 0.763i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.646 - 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(270\)    =    \(2 \cdot 3^{3} \cdot 5\)
Sign: $0.646 - 0.763i$
Analytic conductor: \(171.546\)
Root analytic conductor: \(13.0975\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{270} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 270,\ (\ :5),\ 0.646 - 0.763i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(1.522126792\)
\(L(\frac12)\) \(\approx\) \(1.522126792\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-19.5 + 11.3i)T \)
3 \( 1 \)
5 \( 1 + (1.21e3 + 698. i)T \)
good7 \( 1 + (5.11e3 + 8.85e3i)T + (-1.41e8 + 2.44e8i)T^{2} \)
11 \( 1 + (-8.38e4 + 4.84e4i)T + (1.29e10 - 2.24e10i)T^{2} \)
13 \( 1 + (5.74e4 - 9.95e4i)T + (-6.89e10 - 1.19e11i)T^{2} \)
17 \( 1 - 1.26e6iT - 2.01e12T^{2} \)
19 \( 1 - 1.73e6T + 6.13e12T^{2} \)
23 \( 1 + (6.63e6 + 3.83e6i)T + (2.07e13 + 3.58e13i)T^{2} \)
29 \( 1 + (2.45e7 - 1.41e7i)T + (2.10e14 - 3.64e14i)T^{2} \)
31 \( 1 + (-1.05e6 + 1.81e6i)T + (-4.09e14 - 7.09e14i)T^{2} \)
37 \( 1 + 1.00e8T + 4.80e15T^{2} \)
41 \( 1 + (-1.09e8 - 6.29e7i)T + (6.71e15 + 1.16e16i)T^{2} \)
43 \( 1 + (4.70e7 + 8.14e7i)T + (-1.08e16 + 1.87e16i)T^{2} \)
47 \( 1 + (1.27e8 - 7.37e7i)T + (2.62e16 - 4.55e16i)T^{2} \)
53 \( 1 + 5.19e6iT - 1.74e17T^{2} \)
59 \( 1 + (3.10e8 + 1.79e8i)T + (2.55e17 + 4.42e17i)T^{2} \)
61 \( 1 + (-3.95e8 - 6.85e8i)T + (-3.56e17 + 6.17e17i)T^{2} \)
67 \( 1 + (-5.84e8 + 1.01e9i)T + (-9.11e17 - 1.57e18i)T^{2} \)
71 \( 1 - 2.00e9iT - 3.25e18T^{2} \)
73 \( 1 - 3.30e9T + 4.29e18T^{2} \)
79 \( 1 + (1.40e8 + 2.42e8i)T + (-4.73e18 + 8.19e18i)T^{2} \)
83 \( 1 + (3.27e9 - 1.88e9i)T + (7.75e18 - 1.34e19i)T^{2} \)
89 \( 1 - 6.97e9iT - 3.11e19T^{2} \)
97 \( 1 + (-5.07e9 - 8.79e9i)T + (-3.68e19 + 6.38e19i)T^{2} \)
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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

−10.44485606965822920524548472427, −9.497864093462380778614678168934, −8.398331813803862466378304296566, −7.25628974218689418489666632270, −6.31207003843841014799479314931, −5.23016309795399728067126942580, −4.03096062285579704074077193945, −3.47733055344367372176340717106, −2.00856247006233420795831992472, −0.933185959450153249958061632248, 0.25107139349765992207881872045, 1.89067947249715420808257209372, 3.05979265205714914487761883593, 3.95334107431467815317691067719, 5.14005262313562520846261142557, 6.03030908338642773776831119027, 7.11066812642289508565610157692, 7.84206592470195664153760241685, 9.083302065730280433000035957241, 9.931500283819994255999257495384

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