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Properties

Label 2-2720-8.5-c1-0-31
Degree $2$
Conductor $2720$
Sign $0.450 - 0.892i$
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  + 1.11i·3-s + i·5-s + 3.27·7-s + 1.75·9-s − 0.150i·11-s − 1.29i·13-s − 1.11·15-s − 17-s + 4.68i·19-s + 3.64i·21-s + 6.66·23-s − 25-s + 5.30i·27-s − 4.60i·29-s + 2.35·31-s + ⋯
L(s)  = 1  + 0.643i·3-s + 0.447i·5-s + 1.23·7-s + 0.585·9-s − 0.0453i·11-s − 0.358i·13-s − 0.287·15-s − 0.242·17-s + 1.07i·19-s + 0.796i·21-s + 1.39·23-s − 0.200·25-s + 1.02i·27-s − 0.856i·29-s + 0.423·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.450 - 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.450 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2720\)    =    \(2^{5} \cdot 5 \cdot 17\)
Sign: $0.450 - 0.892i$
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: \((2,\ 2720,\ (\ :1/2),\ 0.450 - 0.892i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.368663763\)
\(L(\frac12)\) \(\approx\) \(2.368663763\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.11iT - 3T^{2} \)
7 \( 1 - 3.27T + 7T^{2} \)
11 \( 1 + 0.150iT - 11T^{2} \)
13 \( 1 + 1.29iT - 13T^{2} \)
19 \( 1 - 4.68iT - 19T^{2} \)
23 \( 1 - 6.66T + 23T^{2} \)
29 \( 1 + 4.60iT - 29T^{2} \)
31 \( 1 - 2.35T + 31T^{2} \)
37 \( 1 + 3.67iT - 37T^{2} \)
41 \( 1 + 1.59T + 41T^{2} \)
43 \( 1 - 2.84iT - 43T^{2} \)
47 \( 1 - 7.68T + 47T^{2} \)
53 \( 1 - 2.83iT - 53T^{2} \)
59 \( 1 - 1.32iT - 59T^{2} \)
61 \( 1 + 6.88iT - 61T^{2} \)
67 \( 1 + 1.32iT - 67T^{2} \)
71 \( 1 - 16.0T + 71T^{2} \)
73 \( 1 + 2.26T + 73T^{2} \)
79 \( 1 + 4.94T + 79T^{2} \)
83 \( 1 + 9.14iT - 83T^{2} \)
89 \( 1 + 11.5T + 89T^{2} \)
97 \( 1 - 2.03T + 97T^{2} \)
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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

−9.012745577726070081101084105821, −8.129387153493918226019205358579, −7.57533742306764485258698825568, −6.74824375865516138825982127921, −5.73667539129365555116799311917, −4.95879840220684255033802213251, −4.27667363220567961799581341520, −3.45119968920818282054964081285, −2.29118376914547427490813838151, −1.19737319366033618059611016687, 0.938157774562284603347315076135, 1.71051178407016401606647436600, 2.73204318048848171519058775971, 4.14022920710240742267523278275, 4.80813234125985484129072529165, 5.42721263204494170439553606021, 6.73552292486618048524031765151, 7.08376404611362459818163101772, 7.953176173635896070905390021278, 8.632279654853985943216404772724

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