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Properties

Label 2-2720-136.101-c1-0-43
Degree $2$
Conductor $2720$
Sign $0.689 - 0.724i$
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  + 3.41·3-s + 5-s + 2.42i·7-s + 8.68·9-s − 3.57·11-s + 2.84i·13-s + 3.41·15-s + (0.785 + 4.04i)17-s + 4.03i·19-s + 8.29i·21-s − 6.96i·23-s + 25-s + 19.4·27-s − 3.63·29-s + 1.77i·31-s + ⋯
L(s)  = 1  + 1.97·3-s + 0.447·5-s + 0.917i·7-s + 2.89·9-s − 1.07·11-s + 0.789i·13-s + 0.882·15-s + (0.190 + 0.981i)17-s + 0.925i·19-s + 1.81i·21-s − 1.45i·23-s + 0.200·25-s + 3.73·27-s − 0.675·29-s + 0.318i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.689 - 0.724i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.689 - 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2720\)    =    \(2^{5} \cdot 5 \cdot 17\)
Sign: $0.689 - 0.724i$
Analytic conductor: \(21.7193\)
Root analytic conductor: \(4.66039\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2720} (2481, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2720,\ (\ :1/2),\ 0.689 - 0.724i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.056166474\)
\(L(\frac12)\) \(\approx\) \(4.056166474\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
17 \( 1 + (-0.785 - 4.04i)T \)
good3 \( 1 - 3.41T + 3T^{2} \)
7 \( 1 - 2.42iT - 7T^{2} \)
11 \( 1 + 3.57T + 11T^{2} \)
13 \( 1 - 2.84iT - 13T^{2} \)
19 \( 1 - 4.03iT - 19T^{2} \)
23 \( 1 + 6.96iT - 23T^{2} \)
29 \( 1 + 3.63T + 29T^{2} \)
31 \( 1 - 1.77iT - 31T^{2} \)
37 \( 1 - 2.43T + 37T^{2} \)
41 \( 1 - 5.61iT - 41T^{2} \)
43 \( 1 + 11.0iT - 43T^{2} \)
47 \( 1 + 5.57T + 47T^{2} \)
53 \( 1 - 6.19iT - 53T^{2} \)
59 \( 1 + 5.08iT - 59T^{2} \)
61 \( 1 - 5.87T + 61T^{2} \)
67 \( 1 - 3.32iT - 67T^{2} \)
71 \( 1 + 9.10iT - 71T^{2} \)
73 \( 1 + 4.57iT - 73T^{2} \)
79 \( 1 + 14.9iT - 79T^{2} \)
83 \( 1 - 7.53iT - 83T^{2} \)
89 \( 1 + 5.67T + 89T^{2} \)
97 \( 1 + 2.50iT - 97T^{2} \)
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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

−8.801580066805867927228524286208, −8.318777428615055024023787311413, −7.73592093049568379275127419969, −6.80480547155023253692142621044, −5.92683758711041499627777895141, −4.83779488950277175039955651591, −3.95629052622613723328754258789, −3.06210458240004675700687516846, −2.26539775763952892964236710704, −1.73030676191247502335513606550, 1.03884481831961777166909404868, 2.26645814971500730986841943902, 2.95515242443506919426456878839, 3.65589446975791868212034112973, 4.63667339566457249324598765791, 5.46561255521788656375544940230, 6.90702113144547675339429077096, 7.46206412582857781984493701410, 7.904726286055061416227291320480, 8.685215545965205447780898493188

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