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Properties

Label 2-2160-240.107-c0-0-0
Degree $2$
Conductor $2160$
Sign $-0.584 + 0.811i$
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)2-s + 1.00i·4-s + (−0.707 + 0.707i)5-s + (0.707 − 0.707i)8-s + 1.00·10-s + (−0.707 − 0.707i)11-s − i·13-s − 1.00·16-s + (−0.707 + 0.707i)17-s + (−0.707 − 0.707i)20-s + 1.00i·22-s + (−0.707 − 0.707i)23-s − 1.00i·25-s + (−0.707 + 0.707i)26-s + (0.707 − 0.707i)29-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + 1.00i·4-s + (−0.707 + 0.707i)5-s + (0.707 − 0.707i)8-s + 1.00·10-s + (−0.707 − 0.707i)11-s − i·13-s − 1.00·16-s + (−0.707 + 0.707i)17-s + (−0.707 − 0.707i)20-s + 1.00i·22-s + (−0.707 − 0.707i)23-s − 1.00i·25-s + (−0.707 + 0.707i)26-s + (0.707 − 0.707i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2160\)    =    \(2^{4} \cdot 3^{3} \cdot 5\)
Sign: $-0.584 + 0.811i$
Analytic conductor: \(1.07798\)
Root analytic conductor: \(1.03825\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2160} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2160,\ (\ :0),\ -0.584 + 0.811i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4226363575\)
\(L(\frac12)\) \(\approx\) \(0.4226363575\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−8.815169709644298952323206884349, −8.257589023780912760526907398741, −7.76627270096701959519447230154, −6.84402281924162255184241859533, −6.02080508406673625062044473748, −4.73286837016023124939932928017, −3.75032821598482199289521544378, −3.03901239264438875328493908012, −2.17243039326751140882828853663, −0.38826245540949152131986365804, 1.35529478392443849270202305131, 2.57723271062309493000018427452, 4.29337718196958449001127094337, 4.67306372519341234469690156285, 5.67446966181710748852418363260, 6.57756054427567159612012094881, 7.51055036073531728977647840050, 7.79276490462752605072063982899, 8.801904447102993677529781969046, 9.343830888583587316326771434233

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