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Properties

Label 2-3120-3120.2027-c0-0-3
Degree $2$
Conductor $3120$
Sign $0.160 + 0.987i$
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  − i·2-s + 3-s − 4-s − 5-s − i·6-s + i·8-s + 9-s + i·10-s + (1 − i)11-s − 12-s + i·13-s − 15-s + 16-s − i·18-s + 20-s + ⋯
L(s)  = 1  − i·2-s + 3-s − 4-s − 5-s − i·6-s + i·8-s + 9-s + i·10-s + (1 − i)11-s − 12-s + i·13-s − 15-s + 16-s − i·18-s + 20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.160 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.160 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3120\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.160 + 0.987i$
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: \((2,\ 3120,\ (\ :0),\ 0.160 + 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.456459148\)
\(L(\frac12)\) \(\approx\) \(1.456459148\)
\(L(1)\) 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 \( 1 - T \)
5 \( 1 + T \)
13 \( 1 - iT \)
good7 \( 1 - iT^{2} \)
11 \( 1 + (-1 + i)T - iT^{2} \)
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 \( 1 - T^{2} \)
43 \( 1 + 2iT - T^{2} \)
47 \( 1 + (-1 + i)T - iT^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-1 - i)T + iT^{2} \)
61 \( 1 + (-1 + i)T - iT^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + 2T + T^{2} \)
89 \( 1 + T^{2} \)
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

−8.776687176123353722212238263303, −8.343949880518306225392212898636, −7.38634780243892122640163491316, −6.66854440494408755958954937102, −5.40154394867433443605871651224, −4.21069995411958420885521408273, −3.93366816382095786081531929886, −3.17139042588095798040397939376, −2.17170066940580122655554201367, −1.03564479339554401652634251308, 1.20593632940762307838510587049, 2.81208293217715443821681792144, 3.76818786688886610595009088032, 4.29763079150830488278735792588, 5.08113007956624787621703923624, 6.26899764977623790350467591071, 7.10816502412062887068883873013, 7.48926677557670281732014615515, 8.275567117596497396055876007645, 8.701917529301567473170302603204

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