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Properties

Label 2-280-8.5-c1-0-7
Degree $2$
Conductor $280$
Sign $0.391 - 0.919i$
Analytic cond. $2.23581$
Root an. cond. $1.49526$
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  + (−0.864 + 1.11i)2-s + 0.903i·3-s + (−0.504 − 1.93i)4-s − i·5-s + (−1.01 − 0.780i)6-s + 7-s + (2.60 + 1.10i)8-s + 2.18·9-s + (1.11 + 0.864i)10-s − 1.29i·11-s + (1.74 − 0.455i)12-s + 4.20i·13-s + (−0.864 + 1.11i)14-s + 0.903·15-s + (−3.49 + 1.95i)16-s + 7.59·17-s + ⋯
L(s)  = 1  + (−0.611 + 0.791i)2-s + 0.521i·3-s + (−0.252 − 0.967i)4-s − 0.447i·5-s + (−0.412 − 0.318i)6-s + 0.377·7-s + (0.919 + 0.391i)8-s + 0.728·9-s + (0.353 + 0.273i)10-s − 0.389i·11-s + (0.504 − 0.131i)12-s + 1.16i·13-s + (−0.231 + 0.299i)14-s + 0.233·15-s + (−0.872 + 0.488i)16-s + 1.84·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.391 - 0.919i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.391 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.391 - 0.919i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.391 - 0.919i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.864599 + 0.571463i\)
\(L(\frac12)\) \(\approx\) \(0.864599 + 0.571463i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.864 - 1.11i)T \)
5 \( 1 + iT \)
7 \( 1 - T \)
good3 \( 1 - 0.903iT - 3T^{2} \)
11 \( 1 + 1.29iT - 11T^{2} \)
13 \( 1 - 4.20iT - 13T^{2} \)
17 \( 1 - 7.59T + 17T^{2} \)
19 \( 1 - 1.10iT - 19T^{2} \)
23 \( 1 + 2.12T + 23T^{2} \)
29 \( 1 + 1.25iT - 29T^{2} \)
31 \( 1 + 3.58T + 31T^{2} \)
37 \( 1 + 1.87iT - 37T^{2} \)
41 \( 1 - 4.82T + 41T^{2} \)
43 \( 1 - 11.3iT - 43T^{2} \)
47 \( 1 - 8.05T + 47T^{2} \)
53 \( 1 + 10.6iT - 53T^{2} \)
59 \( 1 + 10.5iT - 59T^{2} \)
61 \( 1 + 3.38iT - 61T^{2} \)
67 \( 1 + 9.24iT - 67T^{2} \)
71 \( 1 + 13.1T + 71T^{2} \)
73 \( 1 + 11.1T + 73T^{2} \)
79 \( 1 + 6.94T + 79T^{2} \)
83 \( 1 - 5.74iT - 83T^{2} \)
89 \( 1 - 0.691T + 89T^{2} \)
97 \( 1 - 6.79T + 97T^{2} \)
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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

−11.89147403073419013692418636884, −10.82944006883662566730866109762, −9.836397734362195359574627787680, −9.277262334136142419689820633324, −8.146499722223050601242309471208, −7.34574350424008674342172772067, −6.06250984111033839493136643185, −5.03131581416348779860591573718, −3.98654782335240819792487111305, −1.47396468530327434463412415314, 1.26070055580497924084384456772, 2.75400753381642009640354198471, 4.06300313455597324187930460989, 5.63738286800941299559647814206, 7.38898409801637167777262126634, 7.59638847064810544938184461786, 8.878659798985563426680595275088, 10.21149468599567347947125745911, 10.41001369071253652153649393408, 11.81084968987693141464261991638

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