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Properties

Label 2-1080-9.7-c1-0-11
Degree $2$
Conductor $1080$
Sign $-0.00865 + 0.999i$
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  + (0.5 + 0.866i)5-s + (2.62 − 4.54i)7-s + (1.33 − 2.31i)11-s + (−1.90 − 3.30i)13-s − 3.52·17-s − 4.67·19-s + (−2.47 − 4.29i)23-s + (−0.499 + 0.866i)25-s + (−0.928 + 1.60i)29-s + (4.33 + 7.51i)31-s + 5.24·35-s − 2.67·37-s + (−1.83 − 3.18i)41-s + (−1.76 + 3.05i)43-s + (4.63 − 8.02i)47-s + ⋯
L(s)  = 1  + (0.223 + 0.387i)5-s + (0.991 − 1.71i)7-s + (0.402 − 0.697i)11-s + (−0.529 − 0.916i)13-s − 0.855·17-s − 1.07·19-s + (−0.516 − 0.895i)23-s + (−0.0999 + 0.173i)25-s + (−0.172 + 0.298i)29-s + (0.778 + 1.34i)31-s + 0.886·35-s − 0.439·37-s + (−0.286 − 0.496i)41-s + (−0.269 + 0.466i)43-s + (0.675 − 1.17i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00865 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00865 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $-0.00865 + 0.999i$
Analytic conductor: \(8.62384\)
Root analytic conductor: \(2.93663\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1080} (721, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1080,\ (\ :1/2),\ -0.00865 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.552512357\)
\(L(\frac12)\) \(\approx\) \(1.552512357\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
good7 \( 1 + (-2.62 + 4.54i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.33 + 2.31i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.90 + 3.30i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 3.52T + 17T^{2} \)
19 \( 1 + 4.67T + 19T^{2} \)
23 \( 1 + (2.47 + 4.29i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.928 - 1.60i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.33 - 7.51i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 2.67T + 37T^{2} \)
41 \( 1 + (1.83 + 3.18i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.76 - 3.05i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.63 + 8.02i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 2.85T + 53T^{2} \)
59 \( 1 + (-2.10 - 3.63i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.98 + 6.89i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.429 - 0.744i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 15.1T + 71T^{2} \)
73 \( 1 - 6.28T + 73T^{2} \)
79 \( 1 + (2.81 - 4.87i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.94 - 3.37i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 11T + 89T^{2} \)
97 \( 1 + (-1.91 + 3.32i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.958914059438020272703170364085, −8.583512897377544987936860039769, −8.123965023246725623638197850657, −7.03526785030661545637018693913, −6.56629044776662851252784532334, −5.23631746783155310263870722125, −4.37106135527871424871823460186, −3.50616422195252876109324100465, −2.09930460299801887100033127262, −0.67637749501245849817071905456, 1.89716432056364498896232321471, 2.31805662015664601990052223648, 4.20510548396639863344744533805, 4.85793484892526791064515441928, 5.80298900788317701469554727957, 6.59083341895703847779813870040, 7.77438753777860428181116036989, 8.571695582296492256882070353749, 9.202089730118702905781497307028, 9.806137695679478987904170047923

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