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Properties

Label 2-270-45.7-c2-0-3
Degree $2$
Conductor $270$
Sign $0.373 - 0.927i$
Analytic cond. $7.35696$
Root an. cond. $2.71237$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.36 + 0.366i)2-s + (1.73 − i)4-s + (3.46 + 3.60i)5-s + (1.76 − 0.473i)7-s + (−1.99 + 2i)8-s + (−6.05 − 3.65i)10-s + (−3.86 + 6.69i)11-s + (15.7 + 4.21i)13-s + (−2.24 + 1.29i)14-s + (1.99 − 3.46i)16-s + (−6.97 − 6.97i)17-s + 0.474i·19-s + (9.60 + 2.77i)20-s + (2.82 − 10.5i)22-s + (−18.8 − 5.04i)23-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.433 − 0.250i)4-s + (0.693 + 0.720i)5-s + (0.252 − 0.0677i)7-s + (−0.249 + 0.250i)8-s + (−0.605 − 0.365i)10-s + (−0.351 + 0.608i)11-s + (1.21 + 0.324i)13-s + (−0.160 + 0.0924i)14-s + (0.124 − 0.216i)16-s + (−0.410 − 0.410i)17-s + 0.0249i·19-s + (0.480 + 0.138i)20-s + (0.128 − 0.479i)22-s + (−0.818 − 0.219i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.373 - 0.927i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.373 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(270\)    =    \(2 \cdot 3^{3} \cdot 5\)
Sign: $0.373 - 0.927i$
Analytic conductor: \(7.35696\)
Root analytic conductor: \(2.71237\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{270} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 270,\ (\ :1),\ 0.373 - 0.927i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.08581 + 0.733752i\)
\(L(\frac12)\) \(\approx\) \(1.08581 + 0.733752i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 - 0.366i)T \)
3 \( 1 \)
5 \( 1 + (-3.46 - 3.60i)T \)
good7 \( 1 + (-1.76 + 0.473i)T + (42.4 - 24.5i)T^{2} \)
11 \( 1 + (3.86 - 6.69i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (-15.7 - 4.21i)T + (146. + 84.5i)T^{2} \)
17 \( 1 + (6.97 + 6.97i)T + 289iT^{2} \)
19 \( 1 - 0.474iT - 361T^{2} \)
23 \( 1 + (18.8 + 5.04i)T + (458. + 264.5i)T^{2} \)
29 \( 1 + (-46.3 - 26.7i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-29.0 - 50.3i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (12.4 + 12.4i)T + 1.36e3iT^{2} \)
41 \( 1 + (-20.0 - 34.8i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-7.61 - 28.4i)T + (-1.60e3 + 924.5i)T^{2} \)
47 \( 1 + (10.6 - 2.84i)T + (1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (-34.6 + 34.6i)T - 2.80e3iT^{2} \)
59 \( 1 + (71.5 - 41.3i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-41.6 + 72.1i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-11.0 + 41.3i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 - 98.7T + 5.04e3T^{2} \)
73 \( 1 + (-10.9 + 10.9i)T - 5.32e3iT^{2} \)
79 \( 1 + (85.9 + 49.6i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (36.0 + 134. i)T + (-5.96e3 + 3.44e3i)T^{2} \)
89 \( 1 + 131. iT - 7.92e3T^{2} \)
97 \( 1 + (52.9 - 14.1i)T + (8.14e3 - 4.70e3i)T^{2} \)
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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

−11.59912645443530867238443142056, −10.68550123717374105609411799696, −10.07877989563663751614224516278, −9.014766682981841869807733724254, −8.079323418606186413491820059323, −6.86004782672289305506079330827, −6.21579466294902443292100703958, −4.79921611967717761732989095732, −2.99309920798821888834256977186, −1.58099926654382195435685834461, 0.907821978983512692349265566814, 2.38771394357755603428806494261, 4.09690461390570769162128168524, 5.63063543892031059014463515507, 6.41770112769859093145024030004, 8.147332717771907423405902019510, 8.474417142276500796777612324649, 9.637365014706411008196009911487, 10.45501501556835089296524396226, 11.38962877805855606243400531079

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