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Properties

Label 2-2160-15.14-c0-0-4
Degree $2$
Conductor $2160$
Sign $0.707 + 0.707i$
Analytic cond. $1.07798$
Root an. cond. $1.03825$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 0.707i)5-s + i·7-s − 1.41i·11-s − i·13-s − 19-s + 1.41·23-s − 1.00i·25-s − 1.41i·29-s + (0.707 + 0.707i)35-s + i·37-s + 1.41i·41-s + 1.41·53-s + (−1.00 − 1.00i)55-s − 61-s + (−0.707 − 0.707i)65-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)5-s + i·7-s − 1.41i·11-s − i·13-s − 19-s + 1.41·23-s − 1.00i·25-s − 1.41i·29-s + (0.707 + 0.707i)35-s + i·37-s + 1.41i·41-s + 1.41·53-s + (−1.00 − 1.00i)55-s − 61-s + (−0.707 − 0.707i)65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2160\)    =    \(2^{4} \cdot 3^{3} \cdot 5\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(1.07798\)
Root analytic conductor: \(1.03825\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2160} (1889, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2160,\ (\ :0),\ 0.707 + 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.307766734\)
\(L(\frac12)\) \(\approx\) \(1.307766734\)
\(L(1)\) 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 \( 1 + (-0.707 + 0.707i)T \)
good7 \( 1 - iT - T^{2} \)
11 \( 1 + 1.41iT - T^{2} \)
13 \( 1 + iT - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 - 1.41T + T^{2} \)
29 \( 1 + 1.41iT - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - iT - T^{2} \)
41 \( 1 - 1.41iT - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - 1.41T + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 - iT - T^{2} \)
71 \( 1 + 1.41iT - T^{2} \)
73 \( 1 - iT - T^{2} \)
79 \( 1 - T + T^{2} \)
83 \( 1 + 1.41T + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - iT - T^{2} \)
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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

−8.981005944189793950412343155257, −8.527560263549555993759251617078, −7.937389993510061633114299284302, −6.54434918050619773763203943153, −5.89068762861157760291461271730, −5.38412833681878083567992811931, −4.46596650223500158213644188886, −3.13525931027800871266299781335, −2.40825720303266386130029002400, −0.994737123430730438835313106031, 1.58062512804029687945387432507, 2.43612157846312220841865952203, 3.68493575279538663600998276755, 4.47748150253517346951882482002, 5.33888141925683638682176174584, 6.50347751916732354917210509194, 7.09867980042917947804189051546, 7.38168266003385136569102654178, 8.859292724376237293166252724572, 9.308065633744256526859794573429

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