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Properties

Label 2-270-135.113-c1-0-6
Degree $2$
Conductor $270$
Sign $0.188 - 0.982i$
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  + (0.996 + 0.0871i)2-s + (0.600 + 1.62i)3-s + (0.984 + 0.173i)4-s + (−0.702 + 2.12i)5-s + (0.456 + 1.67i)6-s + (−0.580 − 0.406i)7-s + (0.965 + 0.258i)8-s + (−2.27 + 1.95i)9-s + (−0.885 + 2.05i)10-s + (0.183 − 0.502i)11-s + (0.309 + 1.70i)12-s + (−0.260 − 2.97i)13-s + (−0.542 − 0.455i)14-s + (−3.87 + 0.132i)15-s + (0.939 + 0.342i)16-s + (1.53 − 0.410i)17-s + ⋯
L(s)  = 1  + (0.704 + 0.0616i)2-s + (0.346 + 0.937i)3-s + (0.492 + 0.0868i)4-s + (−0.314 + 0.949i)5-s + (0.186 + 0.682i)6-s + (−0.219 − 0.153i)7-s + (0.341 + 0.0915i)8-s + (−0.759 + 0.650i)9-s + (−0.279 + 0.649i)10-s + (0.0551 − 0.151i)11-s + (0.0893 + 0.491i)12-s + (−0.0722 − 0.825i)13-s + (−0.145 − 0.121i)14-s + (−0.999 + 0.0342i)15-s + (0.234 + 0.0855i)16-s + (0.371 − 0.0995i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.188 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.188 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(270\)    =    \(2 \cdot 3^{3} \cdot 5\)
Sign: $0.188 - 0.982i$
Analytic conductor: \(2.15596\)
Root analytic conductor: \(1.46831\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{270} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 270,\ (\ :1/2),\ 0.188 - 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49209 + 1.23343i\)
\(L(\frac12)\) \(\approx\) \(1.49209 + 1.23343i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.996 - 0.0871i)T \)
3 \( 1 + (-0.600 - 1.62i)T \)
5 \( 1 + (0.702 - 2.12i)T \)
good7 \( 1 + (0.580 + 0.406i)T + (2.39 + 6.57i)T^{2} \)
11 \( 1 + (-0.183 + 0.502i)T + (-8.42 - 7.07i)T^{2} \)
13 \( 1 + (0.260 + 2.97i)T + (-12.8 + 2.25i)T^{2} \)
17 \( 1 + (-1.53 + 0.410i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-4.21 + 2.43i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.34 - 4.77i)T + (-7.86 + 21.6i)T^{2} \)
29 \( 1 + (-3.54 + 2.97i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-0.226 + 1.28i)T + (-29.1 - 10.6i)T^{2} \)
37 \( 1 + (-1.90 - 7.10i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-3.75 + 4.47i)T + (-7.11 - 40.3i)T^{2} \)
43 \( 1 + (3.29 + 7.06i)T + (-27.6 + 32.9i)T^{2} \)
47 \( 1 + (6.41 - 9.15i)T + (-16.0 - 44.1i)T^{2} \)
53 \( 1 + (-4.01 + 4.01i)T - 53iT^{2} \)
59 \( 1 + (6.74 - 2.45i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (0.669 + 3.79i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (15.3 - 1.34i)T + (65.9 - 11.6i)T^{2} \)
71 \( 1 + (-2.33 - 1.35i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.25 + 12.1i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (2.08 + 2.48i)T + (-13.7 + 77.7i)T^{2} \)
83 \( 1 + (1.15 - 13.1i)T + (-81.7 - 14.4i)T^{2} \)
89 \( 1 + (7.06 + 12.2i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.89 - 2.74i)T + (62.3 - 74.3i)T^{2} \)
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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

−11.93064817315513257294981552245, −11.19085327738106011979394096540, −10.33862221080146851557648202987, −9.533694042040302811843504708423, −8.103027503000509218352518964169, −7.21482731856450691196426938608, −5.91843961825933040993146446642, −4.84586690445199125714318446181, −3.50887998856802138922063429568, −2.87765237049932738613520995638, 1.40987515653964981267377994434, 3.02183275317500689010303473255, 4.41074173486676797831988047305, 5.62520746800217833674029608010, 6.72668646089757787791079865078, 7.71602093205681142602789187319, 8.701098977049143247763416228582, 9.643075006656895983955594487123, 11.21195888695071731081297308066, 12.13890800173478494890306156726

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