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Properties

Label 2-2160-240.149-c0-0-6
Degree $2$
Conductor $2160$
Sign $-0.382 + 0.923i$
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  − i·2-s − 4-s + (0.707 − 0.707i)5-s + 1.41·7-s + i·8-s + (−0.707 − 0.707i)10-s + (−0.707 + 0.707i)11-s + (−0.707 − 0.707i)13-s − 1.41i·14-s + 16-s − 17-s + (1 − i)19-s + (−0.707 + 0.707i)20-s + (0.707 + 0.707i)22-s − i·23-s + ⋯
L(s)  = 1  − i·2-s − 4-s + (0.707 − 0.707i)5-s + 1.41·7-s + i·8-s + (−0.707 − 0.707i)10-s + (−0.707 + 0.707i)11-s + (−0.707 − 0.707i)13-s − 1.41i·14-s + 16-s − 17-s + (1 − i)19-s + (−0.707 + 0.707i)20-s + (0.707 + 0.707i)22-s − i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2160\)    =    \(2^{4} \cdot 3^{3} \cdot 5\)
Sign: $-0.382 + 0.923i$
Analytic conductor: \(1.07798\)
Root analytic conductor: \(1.03825\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2160} (1349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2160,\ (\ :0),\ -0.382 + 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.292145568\)
\(L(\frac12)\) \(\approx\) \(1.292145568\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 \)
5 \( 1 + (-0.707 + 0.707i)T \)
good7 \( 1 - 1.41T + T^{2} \)
11 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
13 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
17 \( 1 + T + T^{2} \)
19 \( 1 + (-1 + i)T - iT^{2} \)
23 \( 1 + iT - T^{2} \)
29 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
31 \( 1 - T + T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
47 \( 1 - T + T^{2} \)
53 \( 1 + (1 + i)T + iT^{2} \)
59 \( 1 - iT^{2} \)
61 \( 1 - iT^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 - 1.41T + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T + T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + 1.41T + T^{2} \)
97 \( 1 - 1.41iT - T^{2} \)
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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

−9.154680313907273444776197842452, −8.323644875006776803554879314842, −7.88659554238131427615156143458, −6.68243655666069059793955408899, −5.21723281758893574486414839805, −4.99390085124004374958723872221, −4.40886999868329333792093800180, −2.75183510207255080940771329695, −2.16327285279202566813070696185, −1.01710712023317427930031504358, 1.57335299944060337354564257844, 2.78365700511872524726253274281, 4.05802977341906386499302823398, 4.99282014844883649316989246006, 5.55978910583392285385163396250, 6.37868536208954240287024296718, 7.23872271225403484937018266483, 7.87386351484362660668228086988, 8.500218943081132870087816643589, 9.442239537999100904886356760569

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