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Properties

Label 2-1080-40.29-c1-0-79
Degree $2$
Conductor $1080$
Sign $-0.468 + 0.883i$
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  + (0.986 − 1.01i)2-s + (−0.0536 − 1.99i)4-s + (0.569 + 2.16i)5-s − 4.64i·7-s + (−2.07 − 1.91i)8-s + (2.75 + 1.55i)10-s + 3.44i·11-s + 2.72·13-s + (−4.70 − 4.58i)14-s + (−3.99 + 0.214i)16-s − 2.43i·17-s − 7.45i·19-s + (4.29 − 1.25i)20-s + (3.49 + 3.40i)22-s − 6.60i·23-s + ⋯
L(s)  = 1  + (0.697 − 0.716i)2-s + (−0.0268 − 0.999i)4-s + (0.254 + 0.967i)5-s − 1.75i·7-s + (−0.734 − 0.678i)8-s + (0.870 + 0.492i)10-s + 1.04i·11-s + 0.756·13-s + (−1.25 − 1.22i)14-s + (−0.998 + 0.0536i)16-s − 0.589i·17-s − 1.71i·19-s + (0.959 − 0.280i)20-s + (0.745 + 0.725i)22-s − 1.37i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.468 + 0.883i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.468 + 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $-0.468 + 0.883i$
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: \((2,\ 1080,\ (\ :1/2),\ -0.468 + 0.883i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.266072905\)
\(L(\frac12)\) \(\approx\) \(2.266072905\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.986 + 1.01i)T \)
3 \( 1 \)
5 \( 1 + (-0.569 - 2.16i)T \)
good7 \( 1 + 4.64iT - 7T^{2} \)
11 \( 1 - 3.44iT - 11T^{2} \)
13 \( 1 - 2.72T + 13T^{2} \)
17 \( 1 + 2.43iT - 17T^{2} \)
19 \( 1 + 7.45iT - 19T^{2} \)
23 \( 1 + 6.60iT - 23T^{2} \)
29 \( 1 - 0.952iT - 29T^{2} \)
31 \( 1 - 5.77T + 31T^{2} \)
37 \( 1 - 0.0145T + 37T^{2} \)
41 \( 1 - 4.42T + 41T^{2} \)
43 \( 1 + 6.83T + 43T^{2} \)
47 \( 1 - 6.05iT - 47T^{2} \)
53 \( 1 - 4.54T + 53T^{2} \)
59 \( 1 + 9.70iT - 59T^{2} \)
61 \( 1 - 1.57iT - 61T^{2} \)
67 \( 1 - 7.29T + 67T^{2} \)
71 \( 1 + 12.8T + 71T^{2} \)
73 \( 1 + 3.38iT - 73T^{2} \)
79 \( 1 - 5.03T + 79T^{2} \)
83 \( 1 - 12.7T + 83T^{2} \)
89 \( 1 - 0.535T + 89T^{2} \)
97 \( 1 + 9.22iT - 97T^{2} \)
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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

−10.00351627483382211870141108363, −9.143220262962084618458575879533, −7.67508299480818245173859946184, −6.75495594086074973906664823082, −6.50509505042952646997252849723, −4.89794818552085207403268675617, −4.29668942540546666280135881822, −3.29033559405318836956262243021, −2.30728891959402793361073516807, −0.827066413124013342261948248752, 1.77099571080074790810869295223, 3.13878038458127269086634143825, 4.08553865898097462520156110823, 5.43740927065971873646369658222, 5.70417083899719692950401066884, 6.33165033104872582202600301060, 7.937956240825355115212737370363, 8.442436103435682205412805857417, 8.924703416730240010741913845971, 9.894350391812163457179034169635

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