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Properties

Label 2-1080-40.37-c0-0-1
Degree $2$
Conductor $1080$
Sign $0.685 - 0.727i$
Analytic cond. $0.538990$
Root an. cond. $0.734159$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 0.707i)2-s − 1.00i·4-s + (0.965 − 0.258i)5-s + (−0.366 − 0.366i)7-s + (0.707 + 0.707i)8-s + (−0.500 + 0.866i)10-s + 1.93i·11-s + 0.517·14-s − 1.00·16-s + (−0.258 − 0.965i)20-s + (−1.36 − 1.36i)22-s + (0.866 − 0.499i)25-s + (−0.366 + 0.366i)28-s + 1.41·29-s + 1.73·31-s + (0.707 − 0.707i)32-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s − 1.00i·4-s + (0.965 − 0.258i)5-s + (−0.366 − 0.366i)7-s + (0.707 + 0.707i)8-s + (−0.500 + 0.866i)10-s + 1.93i·11-s + 0.517·14-s − 1.00·16-s + (−0.258 − 0.965i)20-s + (−1.36 − 1.36i)22-s + (0.866 − 0.499i)25-s + (−0.366 + 0.366i)28-s + 1.41·29-s + 1.73·31-s + (0.707 − 0.707i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.685 - 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.685 - 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $0.685 - 0.727i$
Analytic conductor: \(0.538990\)
Root analytic conductor: \(0.734159\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1080} (757, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1080,\ (\ :0),\ 0.685 - 0.727i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8433655929\)
\(L(\frac12)\) \(\approx\) \(0.8433655929\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 \)
5 \( 1 + (-0.965 + 0.258i)T \)
good7 \( 1 + (0.366 + 0.366i)T + iT^{2} \)
11 \( 1 - 1.93iT - T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 - 1.41T + T^{2} \)
31 \( 1 - 1.73T + T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (1.22 + 1.22i)T + iT^{2} \)
59 \( 1 + 1.41T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (1.36 - 1.36i)T - iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-1.36 - 1.36i)T + iT^{2} \)
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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

−10.08378262147486221844684903878, −9.473209717263361815288544714471, −8.586998725796640518753789461033, −7.66330508147666693254635133218, −6.75551505075748372338913150325, −6.30209685118638907055192062779, −5.05306015253592566678220273302, −4.49267280279601855501802846462, −2.53641185146829211674990055945, −1.42724407363392269693450952963, 1.19099307325970829387789431519, 2.73017539368283303716443561325, 3.17082215132204956670950282592, 4.65652896867987921025858167688, 6.02325624471988256209836571733, 6.43641899188319161996324195913, 7.76601605085656480403735286573, 8.633206942077530261030524986235, 9.131067954554334293200318244211, 10.05807991073016055280784215812

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