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Properties

Label 2-280-280.59-c1-0-15
Degree $2$
Conductor $280$
Sign $0.937 + 0.349i$
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  + (−0.707 − 1.22i)2-s + (−0.999 + 1.73i)4-s + (1.93 − 1.11i)5-s + (1.41 + 2.23i)7-s + 2.82·8-s + (1.5 + 2.59i)9-s + (−2.73 − 1.58i)10-s + (−3.23 + 5.60i)11-s − 2.66i·13-s + (1.73 − 3.31i)14-s + (−2.00 − 3.46i)16-s + (2.12 − 3.67i)18-s + (7.23 − 4.17i)19-s + 4.47i·20-s + 9.16·22-s + (0.184 + 0.320i)23-s + ⋯
L(s)  = 1  + (−0.499 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.866 − 0.499i)5-s + (0.534 + 0.845i)7-s + 0.999·8-s + (0.5 + 0.866i)9-s + (−0.866 − 0.499i)10-s + (−0.976 + 1.69i)11-s − 0.738i·13-s + (0.464 − 0.885i)14-s + (−0.500 − 0.866i)16-s + (0.499 − 0.866i)18-s + (1.66 − 0.958i)19-s + 0.999i·20-s + 1.95·22-s + (0.0385 + 0.0667i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.937 + 0.349i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.937 + 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.937 + 0.349i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.937 + 0.349i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13513 - 0.204511i\)
\(L(\frac12)\) \(\approx\) \(1.13513 - 0.204511i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 1.22i)T \)
5 \( 1 + (-1.93 + 1.11i)T \)
7 \( 1 + (-1.41 - 2.23i)T \)
good3 \( 1 + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (3.23 - 5.60i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.66iT - 13T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-7.23 + 4.17i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.184 - 0.320i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.76 + 8.25i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 4.59iT - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + (7.93 - 4.57i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.39 + 7.61i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.47 + 3.16i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-10.9 + 6.32i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−11.81647942493383782404243913948, −10.69463195972001508944629566479, −9.900663696995476588089776426188, −9.260296061486866009731445216891, −8.062440765294763493773484269131, −7.32340645169243811254011573738, −5.21034759731477675395818614737, −4.84562906962931722277601769456, −2.66490093073276848254757827511, −1.74241139406730770926300403374, 1.25241925759984226402549138537, 3.49722136184200058934076204008, 5.13934382679463075715741599552, 6.07511984521330905535541810520, 7.01141712568947883643317869367, 7.946031466076859764826849324754, 9.022487834985518767724144126864, 10.01272365782358356556425522413, 10.60771339767991278206191949562, 11.66943062283768494380534193300

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