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Properties

Label 2-270-135.38-c1-0-2
Degree $2$
Conductor $270$
Sign $0.586 - 0.809i$
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  + (−0.0871 − 0.996i)2-s + (1.03 + 1.39i)3-s + (−0.984 + 0.173i)4-s + (−1.59 + 1.57i)5-s + (1.29 − 1.14i)6-s + (0.514 + 0.734i)7-s + (0.258 + 0.965i)8-s + (−0.877 + 2.86i)9-s + (1.70 + 1.44i)10-s + (0.457 + 1.25i)11-s + (−1.25 − 1.19i)12-s + (1.07 + 0.0942i)13-s + (0.687 − 0.576i)14-s + (−3.82 − 0.597i)15-s + (0.939 − 0.342i)16-s + (−0.965 + 3.60i)17-s + ⋯
L(s)  = 1  + (−0.0616 − 0.704i)2-s + (0.594 + 0.803i)3-s + (−0.492 + 0.0868i)4-s + (−0.711 + 0.702i)5-s + (0.529 − 0.468i)6-s + (0.194 + 0.277i)7-s + (0.0915 + 0.341i)8-s + (−0.292 + 0.956i)9-s + (0.538 + 0.458i)10-s + (0.137 + 0.378i)11-s + (−0.362 − 0.344i)12-s + (0.298 + 0.0261i)13-s + (0.183 − 0.154i)14-s + (−0.988 − 0.154i)15-s + (0.234 − 0.0855i)16-s + (−0.234 + 0.873i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.586 - 0.809i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.586 - 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(270\)    =    \(2 \cdot 3^{3} \cdot 5\)
Sign: $0.586 - 0.809i$
Analytic conductor: \(2.15596\)
Root analytic conductor: \(1.46831\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{270} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 270,\ (\ :1/2),\ 0.586 - 0.809i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09321 + 0.557733i\)
\(L(\frac12)\) \(\approx\) \(1.09321 + 0.557733i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0871 + 0.996i)T \)
3 \( 1 + (-1.03 - 1.39i)T \)
5 \( 1 + (1.59 - 1.57i)T \)
good7 \( 1 + (-0.514 - 0.734i)T + (-2.39 + 6.57i)T^{2} \)
11 \( 1 + (-0.457 - 1.25i)T + (-8.42 + 7.07i)T^{2} \)
13 \( 1 + (-1.07 - 0.0942i)T + (12.8 + 2.25i)T^{2} \)
17 \( 1 + (0.965 - 3.60i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-4.47 - 2.58i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.268 + 0.187i)T + (7.86 + 21.6i)T^{2} \)
29 \( 1 + (3.73 + 3.13i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (1.54 + 8.76i)T + (-29.1 + 10.6i)T^{2} \)
37 \( 1 + (-10.2 - 2.74i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-0.945 - 1.12i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (-1.02 - 0.475i)T + (27.6 + 32.9i)T^{2} \)
47 \( 1 + (-6.61 + 4.63i)T + (16.0 - 44.1i)T^{2} \)
53 \( 1 + (-2.08 + 2.08i)T - 53iT^{2} \)
59 \( 1 + (11.7 + 4.26i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-1.05 + 5.97i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (0.832 - 9.51i)T + (-65.9 - 11.6i)T^{2} \)
71 \( 1 + (2.95 - 1.70i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.30 + 1.15i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (10.7 - 12.8i)T + (-13.7 - 77.7i)T^{2} \)
83 \( 1 + (-3.40 + 0.298i)T + (81.7 - 14.4i)T^{2} \)
89 \( 1 + (-4.47 + 7.75i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.89 - 10.5i)T + (-62.3 - 74.3i)T^{2} \)
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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

−11.70472409825238538682873900975, −11.15700004905577788435583120853, −10.17358074898997804472186912664, −9.436977616975100043630757339768, −8.289783576994984235850267579731, −7.56979873420174829148426558298, −5.84332817090261534741773317538, −4.33849408990844967635649232274, −3.58201007017667065152905931293, −2.30313583276525640584410048698, 0.995129609157330650090125696983, 3.23591943739761256677883742549, 4.58112288185936654889371314175, 5.86288479603066339479685743165, 7.24978470953891401346999337014, 7.65044558878661405943114006635, 8.848826958892990928045546674002, 9.256739452240251217116108138160, 11.01610593210790013411205397600, 11.97782399282624372652461640768

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