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Properties

Label 2-1080-9.7-c1-0-8
Degree $2$
Conductor $1080$
Sign $0.254 + 0.967i$
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 + (0.724 − 1.25i)7-s + 2·17-s + 2.89·19-s + (−1.27 − 2.20i)23-s + (−0.499 + 0.866i)25-s + (3.94 − 6.84i)29-s + (−5.44 − 9.43i)31-s − 1.44·35-s − 6·37-s + (−0.0505 − 0.0874i)41-s + (3.89 − 6.75i)43-s + (−2.27 + 3.94i)47-s + (2.44 + 4.24i)49-s + 11.7·53-s + ⋯
L(s)  = 1  + (−0.223 − 0.387i)5-s + (0.273 − 0.474i)7-s + 0.485·17-s + 0.665·19-s + (−0.265 − 0.460i)23-s + (−0.0999 + 0.173i)25-s + (0.733 − 1.27i)29-s + (−0.978 − 1.69i)31-s − 0.245·35-s − 0.986·37-s + (−0.00788 − 0.0136i)41-s + (0.594 − 1.02i)43-s + (−0.331 + 0.574i)47-s + (0.349 + 0.606i)49-s + 1.62·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.254 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.254 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $0.254 + 0.967i$
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: \((2,\ 1080,\ (\ :1/2),\ 0.254 + 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.444484445\)
\(L(\frac12)\) \(\approx\) \(1.444484445\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.724 + 1.25i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 - 2.89T + 19T^{2} \)
23 \( 1 + (1.27 + 2.20i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.94 + 6.84i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.44 + 9.43i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 6T + 37T^{2} \)
41 \( 1 + (0.0505 + 0.0874i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.89 + 6.75i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.27 - 3.94i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 11.7T + 53T^{2} \)
59 \( 1 + (5.44 + 9.43i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.5 + 2.59i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.62 + 9.74i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.79T + 71T^{2} \)
73 \( 1 + 5.79T + 73T^{2} \)
79 \( 1 + (1.44 - 2.51i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.275 - 0.476i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 16.7T + 89T^{2} \)
97 \( 1 + (1 - 1.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.710242288739172369920158646825, −8.897747918828603973893160622943, −7.913311403201780918705754249551, −7.44064611342577388944441532367, −6.27844336761303265277587165661, −5.37742451823023760208550305432, −4.40178002450324256171389852068, −3.56866779480281548400197413893, −2.16308092394424266284706044665, −0.69162234113743948380570937463, 1.45296149554972853303682292832, 2.84867467412086621589150410572, 3.71389910012142942449426314184, 5.01619422898464195837722965890, 5.67113057321228953832014689934, 6.85612417403662923373763528570, 7.46192743168656990907506120313, 8.502359555690684676005289405238, 9.097104343315928637032060107694, 10.20102387281052460426331915407

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