Properties

Label 2-280-1.1-c5-0-5
Degree $2$
Conductor $280$
Sign $1$
Analytic cond. $44.9074$
Root an. cond. $6.70130$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 12·3-s + 25·5-s − 49·7-s − 99·9-s + 556·11-s − 354·13-s − 300·15-s + 770·17-s − 2.68e3·19-s + 588·21-s − 1.52e3·23-s + 625·25-s + 4.10e3·27-s − 2.41e3·29-s + 7.84e3·31-s − 6.67e3·33-s − 1.22e3·35-s − 314·37-s + 4.24e3·39-s − 1.78e4·41-s + 1.64e4·43-s − 2.47e3·45-s + 5.37e3·47-s + 2.40e3·49-s − 9.24e3·51-s + 1.65e3·53-s + 1.39e4·55-s + ⋯
L(s)  = 1  − 0.769·3-s + 0.447·5-s − 0.377·7-s − 0.407·9-s + 1.38·11-s − 0.580·13-s − 0.344·15-s + 0.646·17-s − 1.70·19-s + 0.290·21-s − 0.602·23-s + 1/5·25-s + 1.08·27-s − 0.533·29-s + 1.46·31-s − 1.06·33-s − 0.169·35-s − 0.0377·37-s + 0.447·39-s − 1.66·41-s + 1.35·43-s − 0.182·45-s + 0.354·47-s + 1/7·49-s − 0.497·51-s + 0.0808·53-s + 0.619·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $1$
Analytic conductor: \(44.9074\)
Root analytic conductor: \(6.70130\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: $\chi_{280} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(1.356110606\)
\(L(\frac12)\) \(\approx\) \(1.356110606\)
\(L(\frac{7}{2})\) 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 \( 1 - p^{2} T \)
7 \( 1 + p^{2} T \)
good3 \( 1 + 4 p T + p^{5} T^{2} \)
11 \( 1 - 556 T + p^{5} T^{2} \)
13 \( 1 + 354 T + p^{5} T^{2} \)
17 \( 1 - 770 T + p^{5} T^{2} \)
19 \( 1 + 2684 T + p^{5} T^{2} \)
23 \( 1 + 1528 T + p^{5} T^{2} \)
29 \( 1 + 2418 T + p^{5} T^{2} \)
31 \( 1 - 7840 T + p^{5} T^{2} \)
37 \( 1 + 314 T + p^{5} T^{2} \)
41 \( 1 + 17878 T + p^{5} T^{2} \)
43 \( 1 - 16476 T + p^{5} T^{2} \)
47 \( 1 - 5376 T + p^{5} T^{2} \)
53 \( 1 - 1654 T + p^{5} T^{2} \)
59 \( 1 + 29492 T + p^{5} T^{2} \)
61 \( 1 - 27630 T + p^{5} T^{2} \)
67 \( 1 - 57716 T + p^{5} T^{2} \)
71 \( 1 - 70648 T + p^{5} T^{2} \)
73 \( 1 - 74202 T + p^{5} T^{2} \)
79 \( 1 - 74336 T + p^{5} T^{2} \)
83 \( 1 - 44068 T + p^{5} T^{2} \)
89 \( 1 - 129306 T + p^{5} T^{2} \)
97 \( 1 + 137646 T + p^{5} T^{2} \)
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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

−11.03993691090851066650029010086, −10.12100243545431949172113643143, −9.229925736160324249474859507324, −8.197969635177208048988552475294, −6.64509786481370561227710631321, −6.20198862057844442205319402918, −5.03046683408360222816438448333, −3.77805837456073661989769635275, −2.21961225180144609024350326995, −0.68388353437272910652694347456, 0.68388353437272910652694347456, 2.21961225180144609024350326995, 3.77805837456073661989769635275, 5.03046683408360222816438448333, 6.20198862057844442205319402918, 6.64509786481370561227710631321, 8.197969635177208048988552475294, 9.229925736160324249474859507324, 10.12100243545431949172113643143, 11.03993691090851066650029010086

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