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Properties

Label 2-1080-5.4-c1-0-15
Degree $2$
Conductor $1080$
Sign $0.894 + 0.447i$
Analytic cond. $8.62384$
Root an. cond. $2.93663$
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  + (2 + i)5-s − 2i·7-s + 2·11-s − 2i·13-s − 3i·17-s + 19-s − 5i·23-s + (3 + 4i)25-s + 10·29-s − 9·31-s + (2 − 4i)35-s + 8i·37-s + 2·41-s − 6i·43-s + 3·49-s + ⋯
L(s)  = 1  + (0.894 + 0.447i)5-s − 0.755i·7-s + 0.603·11-s − 0.554i·13-s − 0.727i·17-s + 0.229·19-s − 1.04i·23-s + (0.600 + 0.800i)25-s + 1.85·29-s − 1.61·31-s + (0.338 − 0.676i)35-s + 1.31i·37-s + 0.312·41-s − 0.914i·43-s + 0.428·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(8.62384\)
Root analytic conductor: \(2.93663\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1080} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1080,\ (\ :1/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.965013341\)
\(L(\frac12)\) \(\approx\) \(1.965013341\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (-2 - i)T \)
good7 \( 1 + 2iT - 7T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 + 3iT - 17T^{2} \)
19 \( 1 - T + 19T^{2} \)
23 \( 1 + 5iT - 23T^{2} \)
29 \( 1 - 10T + 29T^{2} \)
31 \( 1 + 9T + 31T^{2} \)
37 \( 1 - 8iT - 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 6iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + iT - 53T^{2} \)
59 \( 1 - 6T + 59T^{2} \)
61 \( 1 + 7T + 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 - 14T + 71T^{2} \)
73 \( 1 - 14iT - 73T^{2} \)
79 \( 1 + T + 79T^{2} \)
83 \( 1 - 5iT - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 - 8iT - 97T^{2} \)
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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

−9.940789225630553580320347951425, −9.108730286714296595999822574521, −8.195870110120559952995869756606, −7.07317666073491225088390869956, −6.61460873040219723125682358289, −5.57181694831655513835739127937, −4.64050777312722659197273945002, −3.46214549499373544602214124159, −2.44081768725416474157069752956, −1.01241866702629085475385088733, 1.40255010553599565288300511069, 2.39076559527693328853245470243, 3.73942135620161958421240801744, 4.86433047262160709736722479948, 5.75524748360216235159367403043, 6.35971509827732581854685535964, 7.42299816622332251201725509227, 8.578112561052292692602497908993, 9.117213590418413797314779771956, 9.741908590729912402692369001677

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