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Properties

Label 2-1080-120.59-c1-0-12
Degree $2$
Conductor $1080$
Sign $0.513 - 0.858i$
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  + (−0.341 − 1.37i)2-s + (−1.76 + 0.936i)4-s + (−2.13 + 0.663i)5-s + 4.39·7-s + (1.88 + 2.10i)8-s + (1.63 + 2.70i)10-s + 0.633i·11-s − 3.40·13-s + (−1.50 − 6.03i)14-s + (2.24 − 3.30i)16-s − 5.48·17-s + 0.323·19-s + (3.15 − 3.17i)20-s + (0.868 − 0.215i)22-s + 6.26i·23-s + ⋯
L(s)  = 1  + (−0.241 − 0.970i)2-s + (−0.883 + 0.468i)4-s + (−0.954 + 0.296i)5-s + 1.66·7-s + (0.667 + 0.744i)8-s + (0.518 + 0.855i)10-s + 0.190i·11-s − 0.944·13-s + (−0.400 − 1.61i)14-s + (0.561 − 0.827i)16-s − 1.33·17-s + 0.0741·19-s + (0.704 − 0.709i)20-s + (0.185 − 0.0460i)22-s + 1.30i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.513 - 0.858i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.513 - 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $0.513 - 0.858i$
Analytic conductor: \(8.62384\)
Root analytic conductor: \(2.93663\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1080} (539, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1080,\ (\ :1/2),\ 0.513 - 0.858i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7152819318\)
\(L(\frac12)\) \(\approx\) \(0.7152819318\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.341 + 1.37i)T \)
3 \( 1 \)
5 \( 1 + (2.13 - 0.663i)T \)
good7 \( 1 - 4.39T + 7T^{2} \)
11 \( 1 - 0.633iT - 11T^{2} \)
13 \( 1 + 3.40T + 13T^{2} \)
17 \( 1 + 5.48T + 17T^{2} \)
19 \( 1 - 0.323T + 19T^{2} \)
23 \( 1 - 6.26iT - 23T^{2} \)
29 \( 1 + 3.77T + 29T^{2} \)
31 \( 1 - 7.86iT - 31T^{2} \)
37 \( 1 + 5.90T + 37T^{2} \)
41 \( 1 + 0.877iT - 41T^{2} \)
43 \( 1 + 2.31iT - 43T^{2} \)
47 \( 1 - 8.67iT - 47T^{2} \)
53 \( 1 - 7.09iT - 53T^{2} \)
59 \( 1 + 10.5iT - 59T^{2} \)
61 \( 1 - 14.0iT - 61T^{2} \)
67 \( 1 - 5.99iT - 67T^{2} \)
71 \( 1 - 10.7T + 71T^{2} \)
73 \( 1 - 13.9iT - 73T^{2} \)
79 \( 1 + 7.33iT - 79T^{2} \)
83 \( 1 + 1.90T + 83T^{2} \)
89 \( 1 + 8.16iT - 89T^{2} \)
97 \( 1 - 6.27iT - 97T^{2} \)
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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

−10.25644955967884538791241590972, −9.115954039983485850626907236840, −8.482055997779479498271028302660, −7.64313154637181020198344959755, −7.12697350792253926120183271033, −5.25469584208182156896162687375, −4.63516616784831436248039758805, −3.80761816574898399746650471729, −2.55030868370253273389891683642, −1.48218413860199487271725080361, 0.36074508134659259598076077555, 2.03418812683035406569488833293, 3.98126903923817715726919297797, 4.71076997858518433639375823081, 5.24378513055120292453229392588, 6.59094315224194145257164781226, 7.40836592144243341984970127424, 8.100972801702444446638804966182, 8.577512728615746987014360212437, 9.424807255861360419465717862250

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