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Properties

Label 2-1080-40.29-c1-0-88
Degree $2$
Conductor $1080$
Sign $-0.914 + 0.405i$
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  + (0.564 − 1.29i)2-s + (−1.36 − 1.46i)4-s + (2.22 − 0.173i)5-s − 3.43i·7-s + (−2.66 + 0.943i)8-s + (1.03 − 2.98i)10-s − 4.54i·11-s + 1.84·13-s + (−4.45 − 1.93i)14-s + (−0.281 + 3.99i)16-s + 0.380i·17-s + 1.23i·19-s + (−3.29 − 3.02i)20-s + (−5.89 − 2.56i)22-s + 5.35i·23-s + ⋯
L(s)  = 1  + (0.398 − 0.917i)2-s + (−0.681 − 0.731i)4-s + (0.996 − 0.0774i)5-s − 1.29i·7-s + (−0.942 + 0.333i)8-s + (0.326 − 0.945i)10-s − 1.37i·11-s + 0.510·13-s + (−1.18 − 0.517i)14-s + (−0.0702 + 0.997i)16-s + 0.0923i·17-s + 0.282i·19-s + (−0.736 − 0.676i)20-s + (−1.25 − 0.546i)22-s + 1.11i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.914 + 0.405i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.914 + 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $-0.914 + 0.405i$
Analytic conductor: \(8.62384\)
Root analytic conductor: \(2.93663\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1080} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1080,\ (\ :1/2),\ -0.914 + 0.405i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.003931403\)
\(L(\frac12)\) \(\approx\) \(2.003931403\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.564 + 1.29i)T \)
3 \( 1 \)
5 \( 1 + (-2.22 + 0.173i)T \)
good7 \( 1 + 3.43iT - 7T^{2} \)
11 \( 1 + 4.54iT - 11T^{2} \)
13 \( 1 - 1.84T + 13T^{2} \)
17 \( 1 - 0.380iT - 17T^{2} \)
19 \( 1 - 1.23iT - 19T^{2} \)
23 \( 1 - 5.35iT - 23T^{2} \)
29 \( 1 + 3.17iT - 29T^{2} \)
31 \( 1 + 6.89T + 31T^{2} \)
37 \( 1 + 6.60T + 37T^{2} \)
41 \( 1 - 10.9T + 41T^{2} \)
43 \( 1 + 7.34T + 43T^{2} \)
47 \( 1 + 9.34iT - 47T^{2} \)
53 \( 1 - 1.25T + 53T^{2} \)
59 \( 1 + 3.34iT - 59T^{2} \)
61 \( 1 + 7.74iT - 61T^{2} \)
67 \( 1 + 13.3T + 67T^{2} \)
71 \( 1 - 11.9T + 71T^{2} \)
73 \( 1 - 13.1iT - 73T^{2} \)
79 \( 1 - 13.9T + 79T^{2} \)
83 \( 1 + 0.240T + 83T^{2} \)
89 \( 1 + 5.46T + 89T^{2} \)
97 \( 1 - 15.7iT - 97T^{2} \)
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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

−9.698595280677114148543499294737, −8.986451494813204874629814139273, −8.051154546536896278525183829097, −6.78310211889074064860201383726, −5.86512239399782840104310452416, −5.22190892666600204680688949341, −3.91239729888669125885156306429, −3.32202932742600389736409869164, −1.87229517222004784781014108920, −0.801918586061035010074760596472, 1.98862608515549418211671561854, 2.99949470291049401828671347305, 4.46922397190181220150618141303, 5.25348608265598381314444013030, 5.98632058355549021923130507926, 6.71957712535277045978312682116, 7.57568412052679253202654030953, 8.804678125969004355748823827600, 9.085650922501519939473333522723, 9.957407423687813205020784901770

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