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Properties

Label 2-2720-8.5-c1-0-41
Degree $2$
Conductor $2720$
Sign $0.707 + 0.707i$
Analytic cond. $21.7193$
Root an. cond. $4.66039$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·3-s − i·5-s + 4·7-s + 2·9-s + 2i·11-s − 3i·13-s − 15-s + 17-s + 5i·19-s − 4i·21-s + 4·23-s − 25-s − 5i·27-s − 9i·29-s + 5·31-s + ⋯
L(s)  = 1  − 0.577i·3-s − 0.447i·5-s + 1.51·7-s + 0.666·9-s + 0.603i·11-s − 0.832i·13-s − 0.258·15-s + 0.242·17-s + 1.14i·19-s − 0.872i·21-s + 0.834·23-s − 0.200·25-s − 0.962i·27-s − 1.67i·29-s + 0.898·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2720\)    =    \(2^{5} \cdot 5 \cdot 17\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(21.7193\)
Root analytic conductor: \(4.66039\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2720} (1361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2720,\ (\ :1/2),\ 0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.515756064\)
\(L(\frac12)\) \(\approx\) \(2.515756064\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + iT \)
17 \( 1 - T \)
good3 \( 1 + iT - 3T^{2} \)
7 \( 1 - 4T + 7T^{2} \)
11 \( 1 - 2iT - 11T^{2} \)
13 \( 1 + 3iT - 13T^{2} \)
19 \( 1 - 5iT - 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 + 9iT - 29T^{2} \)
31 \( 1 - 5T + 31T^{2} \)
37 \( 1 - 10iT - 37T^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 - 10iT - 43T^{2} \)
47 \( 1 + 11T + 47T^{2} \)
53 \( 1 - 5iT - 53T^{2} \)
59 \( 1 + 5iT - 59T^{2} \)
61 \( 1 - iT - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 7T + 71T^{2} \)
73 \( 1 - T + 73T^{2} \)
79 \( 1 + 16T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + 7T + 89T^{2} \)
97 \( 1 + 7T + 97T^{2} \)
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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

−8.226433629917956368763346791519, −8.110602151115576243181346879899, −7.47302035811371398118670862886, −6.46235944248525995197627271823, −5.62327091404468722657439929086, −4.68133216916476495207555612649, −4.28749751017247474700206621369, −2.81942456131792309675528530009, −1.67551000938202249349609132119, −1.07577800312028185911388967408, 1.14779935578444598374888042038, 2.21013389562202689060799910479, 3.36234169919006234900263140792, 4.31734593024163560651074346886, 4.89289938944751641714948808860, 5.63802787188002359429604770530, 6.93050541514131039513184423786, 7.23037823476918319678146340768, 8.296537750865724398203420936282, 8.929826230963942263796267352500

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