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Properties

Label 2-1080-8.5-c1-0-29
Degree $2$
Conductor $1080$
Sign $0.201 - 0.979i$
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  + (1.17 + 0.788i)2-s + (0.757 + 1.85i)4-s − i·5-s + 2.12·7-s + (−0.569 + 2.77i)8-s + (0.788 − 1.17i)10-s + 5.48i·11-s − 4.91i·13-s + (2.49 + 1.67i)14-s + (−2.85 + 2.80i)16-s − 0.235·17-s + 5.23i·19-s + (1.85 − 0.757i)20-s + (−4.31 + 6.43i)22-s + 7.42·23-s + ⋯
L(s)  = 1  + (0.830 + 0.557i)2-s + (0.378 + 0.925i)4-s − 0.447i·5-s + 0.804·7-s + (−0.201 + 0.979i)8-s + (0.249 − 0.371i)10-s + 1.65i·11-s − 1.36i·13-s + (0.667 + 0.448i)14-s + (−0.712 + 0.701i)16-s − 0.0571·17-s + 1.20i·19-s + (0.413 − 0.169i)20-s + (−0.920 + 1.37i)22-s + 1.54·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.201 - 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.201 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $0.201 - 0.979i$
Analytic conductor: \(8.62384\)
Root analytic conductor: \(2.93663\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1080} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1080,\ (\ :1/2),\ 0.201 - 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.825899998\)
\(L(\frac12)\) \(\approx\) \(2.825899998\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.17 - 0.788i)T \)
3 \( 1 \)
5 \( 1 + iT \)
good7 \( 1 - 2.12T + 7T^{2} \)
11 \( 1 - 5.48iT - 11T^{2} \)
13 \( 1 + 4.91iT - 13T^{2} \)
17 \( 1 + 0.235T + 17T^{2} \)
19 \( 1 - 5.23iT - 19T^{2} \)
23 \( 1 - 7.42T + 23T^{2} \)
29 \( 1 - 4.24iT - 29T^{2} \)
31 \( 1 - 5.05T + 31T^{2} \)
37 \( 1 + 2.27iT - 37T^{2} \)
41 \( 1 - 3.26T + 41T^{2} \)
43 \( 1 + 9.90iT - 43T^{2} \)
47 \( 1 + 8.17T + 47T^{2} \)
53 \( 1 - 10.6iT - 53T^{2} \)
59 \( 1 + 0.702iT - 59T^{2} \)
61 \( 1 - 0.319iT - 61T^{2} \)
67 \( 1 + 2.70iT - 67T^{2} \)
71 \( 1 - 7.18T + 71T^{2} \)
73 \( 1 - 9.66T + 73T^{2} \)
79 \( 1 + 10.8T + 79T^{2} \)
83 \( 1 + 8.53iT - 83T^{2} \)
89 \( 1 + 9.90T + 89T^{2} \)
97 \( 1 - 14.8T + 97T^{2} \)
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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

−10.14374782920531900782400220362, −9.030074313234078675872057590801, −8.103467580884249477966916320787, −7.57588279743489298722891683323, −6.72611259263248941402120558757, −5.50181136624858930426943718068, −4.98683543218675973775724136246, −4.16579568079262989926790180269, −2.94826422192331525901566107616, −1.65245854773603476657706815198, 1.07324650724521925343966968431, 2.45886620593028515193231466088, 3.33743028699816346880394317042, 4.47125474094662166918717246940, 5.16513421814556985560656879590, 6.33370299126713897618236410083, 6.81015925158586069439718175551, 8.096760298780937071174175777180, 9.005168446869490156228836230581, 9.806683886835963408487217090788

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