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Properties

Label 2-280-280.213-c1-0-7
Degree $2$
Conductor $280$
Sign $0.814 - 0.580i$
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  + (−1.16 + 0.796i)2-s + (−2.48 + 0.665i)3-s + (0.732 − 1.86i)4-s + (−2.21 − 0.331i)5-s + (2.37 − 2.75i)6-s + (−2.18 − 1.49i)7-s + (0.626 + 2.75i)8-s + (3.13 − 1.80i)9-s + (2.84 − 1.37i)10-s + (1.96 + 1.13i)11-s + (−0.579 + 5.11i)12-s + (−0.266 − 0.266i)13-s + (3.74 + 0.00220i)14-s + (5.71 − 0.648i)15-s + (−2.92 − 2.72i)16-s + (5.62 − 1.50i)17-s + ⋯
L(s)  = 1  + (−0.826 + 0.563i)2-s + (−1.43 + 0.384i)3-s + (0.366 − 0.930i)4-s + (−0.988 − 0.148i)5-s + (0.969 − 1.12i)6-s + (−0.826 − 0.563i)7-s + (0.221 + 0.975i)8-s + (1.04 − 0.603i)9-s + (0.900 − 0.434i)10-s + (0.593 + 0.342i)11-s + (−0.167 + 1.47i)12-s + (−0.0739 − 0.0739i)13-s + (0.999 + 0.000590i)14-s + (1.47 − 0.167i)15-s + (−0.732 − 0.681i)16-s + (1.36 − 0.365i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.814 - 0.580i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.814 - 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.814 - 0.580i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (213, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.814 - 0.580i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.349303 + 0.111839i\)
\(L(\frac12)\) \(\approx\) \(0.349303 + 0.111839i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.16 - 0.796i)T \)
5 \( 1 + (2.21 + 0.331i)T \)
7 \( 1 + (2.18 + 1.49i)T \)
good3 \( 1 + (2.48 - 0.665i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (-1.96 - 1.13i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.266 + 0.266i)T + 13iT^{2} \)
17 \( 1 + (-5.62 + 1.50i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (2.59 - 1.49i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.99 - 7.45i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 8.88T + 29T^{2} \)
31 \( 1 + (1.92 + 1.11i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.654 + 2.44i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 0.631iT - 41T^{2} \)
43 \( 1 + (-7.46 + 7.46i)T - 43iT^{2} \)
47 \( 1 + (-1.07 + 4.01i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (0.645 + 2.41i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-1.38 - 0.797i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.61 + 6.25i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.68 + 0.987i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 7.81T + 71T^{2} \)
73 \( 1 + (-0.809 - 3.02i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-14.7 + 8.54i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.50 - 8.50i)T + 83iT^{2} \)
89 \( 1 + (-3.06 - 5.31i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.08 - 7.08i)T + 97iT^{2} \)
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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

−11.86573366583144920853539478361, −10.82376512515586257934275197388, −10.15377749693327126739427178482, −9.294464072637760653676273349434, −7.87051020458542183090391398376, −7.03290179895127258710897350479, −6.11134125284870599210931817125, −5.08348865501839897856973942819, −3.80792179281918462411747452413, −0.75392543809255217030640524725, 0.76102235114664884127613514169, 2.99522740846261874133133390199, 4.41379091432622820199098912427, 6.15034408492002043887091365723, 6.74298855323848804417701021476, 7.947150097169603196625096780918, 8.927573013519177260568423863283, 10.22545078235089730729017527606, 10.86900818106943031405589544208, 11.88318609295991463452688646325

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