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Properties

Label 2-3120-3120.2027-c0-0-0
Degree $2$
Conductor $3120$
Sign $-0.811 - 0.584i$
Analytic cond. $1.55708$
Root an. cond. $1.24783$
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 − 3-s + 1.00i·4-s + (0.707 − 0.707i)5-s + (−0.707 − 0.707i)6-s + (−0.707 + 0.707i)8-s + 9-s + 1.00·10-s + (−1.41 + 1.41i)11-s − 1.00i·12-s + i·13-s + (−0.707 + 0.707i)15-s − 1.00·16-s + (0.707 + 0.707i)18-s + (0.707 + 0.707i)20-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s − 3-s + 1.00i·4-s + (0.707 − 0.707i)5-s + (−0.707 − 0.707i)6-s + (−0.707 + 0.707i)8-s + 9-s + 1.00·10-s + (−1.41 + 1.41i)11-s − 1.00i·12-s + i·13-s + (−0.707 + 0.707i)15-s − 1.00·16-s + (0.707 + 0.707i)18-s + (0.707 + 0.707i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.811 - 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.811 - 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3120\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.811 - 0.584i$
Analytic conductor: \(1.55708\)
Root analytic conductor: \(1.24783\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3120} (2027, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3120,\ (\ :0),\ -0.811 - 0.584i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.095872569\)
\(L(\frac12)\) \(\approx\) \(1.095872569\)
\(L(1)\) 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 \( 1 + T \)
5 \( 1 + (-0.707 + 0.707i)T \)
13 \( 1 - iT \)
good7 \( 1 - iT^{2} \)
11 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 - iT^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - 1.41iT - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-1.41 - 1.41i)T + iT^{2} \)
61 \( 1 + (1 - i)T - iT^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + 1.41T + T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 - 2T + T^{2} \)
83 \( 1 + 1.41T + T^{2} \)
89 \( 1 + 1.41T + T^{2} \)
97 \( 1 - iT^{2} \)
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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

−9.200929457724873633505725824042, −8.211352780823721080902583785069, −7.39180370344871736593928701480, −6.84385215899514981279633159720, −5.95194702337665885649850113668, −5.42075481276140631105990951949, −4.51485023830849521104471186927, −4.42834666225701263354877117650, −2.68323672571061776546235722993, −1.69998018891399121728414002839, 0.57254054588633867349655608148, 2.01819095587703572824417953366, 2.97236879456095759459790203258, 3.66719852688076828673360534657, 5.00923349997051766756786616710, 5.50316547730226930605964021574, 5.95526759812397872370497110434, 6.74493785198664105891943404250, 7.65506958153628222827609832626, 8.676860392547067699887913061508

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