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Properties

Label 2-280-8.5-c1-0-22
Degree $2$
Conductor $280$
Sign $-0.707 + 0.707i$
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  + (1 − i)2-s − 2i·3-s − 2i·4-s − i·5-s + (−2 − 2i)6-s + 7-s + (−2 − 2i)8-s − 9-s + (−1 − i)10-s + 4i·11-s − 4·12-s + 6i·13-s + (1 − i)14-s − 2·15-s − 4·16-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s − 1.15i·3-s − i·4-s − 0.447i·5-s + (−0.816 − 0.816i)6-s + 0.377·7-s + (−0.707 − 0.707i)8-s − 0.333·9-s + (−0.316 − 0.316i)10-s + 1.20i·11-s − 1.15·12-s + 1.66i·13-s + (0.267 − 0.267i)14-s − 0.516·15-s − 16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.707 + 0.707i$
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: \((2,\ 280,\ (\ :1/2),\ -0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.705387 - 1.70295i\)
\(L(\frac12)\) \(\approx\) \(0.705387 - 1.70295i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + i)T \)
5 \( 1 + iT \)
7 \( 1 - T \)
good3 \( 1 + 2iT - 3T^{2} \)
11 \( 1 - 4iT - 11T^{2} \)
13 \( 1 - 6iT - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 8iT - 19T^{2} \)
23 \( 1 - 8T + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 - 10iT - 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 - 2iT - 53T^{2} \)
59 \( 1 + 4iT - 59T^{2} \)
61 \( 1 + 2iT - 61T^{2} \)
67 \( 1 - 12iT - 67T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 - 4T + 73T^{2} \)
79 \( 1 + 2T + 79T^{2} \)
83 \( 1 - 6iT - 83T^{2} \)
89 \( 1 - 2T + 89T^{2} \)
97 \( 1 + 16T + 97T^{2} \)
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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

−11.73767487249266841927131091493, −11.00254037234164646613706029393, −9.574246572956148022134268869640, −8.846187386583447001354144671029, −7.09706859958992277369656938957, −6.80055759671756746949027070232, −5.11550649019364488951853432169, −4.32555764497868044400610701493, −2.39355991455377212520811583508, −1.36852156903759209297522432387, 3.15102136648621166249677643412, 3.85272602840362160378447277006, 5.29386791472122641693873848612, 5.81488400434806022753942708023, 7.36047886773411951612778996237, 8.252367082791596531678496744475, 9.262673311020243056479973776772, 10.63935191835904951259489965382, 10.98609206501897297052335205385, 12.38376736999930324333143358773

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