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Properties

Label 2-1080-120.59-c1-0-2
Degree $2$
Conductor $1080$
Sign $-0.370 - 0.928i$
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  + (−1.08 − 0.910i)2-s + (0.343 + 1.97i)4-s + (−1.75 − 1.37i)5-s − 1.14·7-s + (1.42 − 2.44i)8-s + (0.649 + 3.09i)10-s + 0.489i·11-s + 6.59·13-s + (1.24 + 1.04i)14-s + (−3.76 + 1.35i)16-s − 4.58·17-s − 6.55·19-s + (2.11 − 3.94i)20-s + (0.445 − 0.529i)22-s − 6.55i·23-s + ⋯
L(s)  = 1  + (−0.765 − 0.643i)2-s + (0.171 + 0.985i)4-s + (−0.787 − 0.616i)5-s − 0.433·7-s + (0.502 − 0.864i)8-s + (0.205 + 0.978i)10-s + 0.147i·11-s + 1.83·13-s + (0.331 + 0.278i)14-s + (−0.941 + 0.338i)16-s − 1.11·17-s − 1.50·19-s + (0.472 − 0.881i)20-s + (0.0949 − 0.112i)22-s − 1.36i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.370 - 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.370 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $-0.370 - 0.928i$
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: \((2,\ 1080,\ (\ :1/2),\ -0.370 - 0.928i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1000200297\)
\(L(\frac12)\) \(\approx\) \(0.1000200297\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.08 + 0.910i)T \)
3 \( 1 \)
5 \( 1 + (1.75 + 1.37i)T \)
good7 \( 1 + 1.14T + 7T^{2} \)
11 \( 1 - 0.489iT - 11T^{2} \)
13 \( 1 - 6.59T + 13T^{2} \)
17 \( 1 + 4.58T + 17T^{2} \)
19 \( 1 + 6.55T + 19T^{2} \)
23 \( 1 + 6.55iT - 23T^{2} \)
29 \( 1 - 0.212T + 29T^{2} \)
31 \( 1 + 0.796iT - 31T^{2} \)
37 \( 1 + 4.78T + 37T^{2} \)
41 \( 1 - 6.42iT - 41T^{2} \)
43 \( 1 - 2.15iT - 43T^{2} \)
47 \( 1 + 1.38iT - 47T^{2} \)
53 \( 1 - 7.06iT - 53T^{2} \)
59 \( 1 - 8.26iT - 59T^{2} \)
61 \( 1 - 7.61iT - 61T^{2} \)
67 \( 1 - 11.2iT - 67T^{2} \)
71 \( 1 + 11.3T + 71T^{2} \)
73 \( 1 + 9.32iT - 73T^{2} \)
79 \( 1 + 6.01iT - 79T^{2} \)
83 \( 1 + 14.6T + 83T^{2} \)
89 \( 1 - 15.7iT - 89T^{2} \)
97 \( 1 + 6.13iT - 97T^{2} \)
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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

−10.30426861123806556754224916376, −8.964126054422384627519294167108, −8.725895176122375942203024658237, −8.040816962344466345918853980756, −6.86913587026076419103259834192, −6.18728258431061346276936674144, −4.44759926634884943445005650478, −3.97172808453658303039864779922, −2.76834259103640889417048238472, −1.36878522348948383656398306699, 0.06016691573122015561733363324, 1.83091497788905149353949006427, 3.37238557107497307975107662486, 4.30489896366253419551492845531, 5.69150072174354208655356638110, 6.53980626424081647700121495130, 6.97419278546394984963698224316, 8.181803249119466607407389009186, 8.567137552327631221576079247784, 9.448525282259077059173282597632

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