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Properties

Label 2-280-280.107-c1-0-36
Degree $2$
Conductor $280$
Sign $-0.265 + 0.964i$
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  + (−1.34 − 0.431i)2-s + (1.52 + 0.409i)3-s + (1.62 + 1.16i)4-s + (−0.934 − 2.03i)5-s + (−1.88 − 1.21i)6-s + (−1.62 − 2.08i)7-s + (−1.69 − 2.26i)8-s + (−0.426 − 0.245i)9-s + (0.382 + 3.13i)10-s + (−1.76 − 3.05i)11-s + (2.01 + 2.44i)12-s + (−1.68 − 1.68i)13-s + (1.28 + 3.51i)14-s + (−0.597 − 3.49i)15-s + (1.29 + 3.78i)16-s + (0.335 − 1.25i)17-s + ⋯
L(s)  = 1  + (−0.952 − 0.305i)2-s + (0.883 + 0.236i)3-s + (0.813 + 0.581i)4-s + (−0.418 − 0.908i)5-s + (−0.768 − 0.494i)6-s + (−0.613 − 0.789i)7-s + (−0.597 − 0.801i)8-s + (−0.142 − 0.0819i)9-s + (0.120 + 0.992i)10-s + (−0.531 − 0.919i)11-s + (0.581 + 0.705i)12-s + (−0.468 − 0.468i)13-s + (0.343 + 0.939i)14-s + (−0.154 − 0.901i)15-s + (0.324 + 0.945i)16-s + (0.0814 − 0.304i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.265 + 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.265 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.265 + 0.964i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.265 + 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.496630 - 0.651720i\)
\(L(\frac12)\) \(\approx\) \(0.496630 - 0.651720i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.34 + 0.431i)T \)
5 \( 1 + (0.934 + 2.03i)T \)
7 \( 1 + (1.62 + 2.08i)T \)
good3 \( 1 + (-1.52 - 0.409i)T + (2.59 + 1.5i)T^{2} \)
11 \( 1 + (1.76 + 3.05i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.68 + 1.68i)T + 13iT^{2} \)
17 \( 1 + (-0.335 + 1.25i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-6.62 - 3.82i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.50 + 0.940i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 5.07T + 29T^{2} \)
31 \( 1 + (2.42 - 1.39i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.340 - 1.27i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 4.50T + 41T^{2} \)
43 \( 1 + (-7.05 + 7.05i)T - 43iT^{2} \)
47 \( 1 + (2.14 + 7.98i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (3.15 - 11.7i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-10.7 + 6.21i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-11.1 - 6.45i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.866 + 3.23i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 2.60iT - 71T^{2} \)
73 \( 1 + (-1.56 - 0.420i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (3.66 - 6.34i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.94 + 4.94i)T - 83iT^{2} \)
89 \( 1 + (4.71 + 2.71i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.73 + 5.73i)T + 97iT^{2} \)
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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

−11.51089288575859817268546530635, −10.33633015684547061830489474687, −9.600896782196091234539708710048, −8.716217811827506228669387423505, −8.020312917975610367927751534032, −7.16617743014495900868590589872, −5.52678038322724436626006208885, −3.72553414756063924488023065430, −2.94442432746198709162547243095, −0.74689188125180242339050277741, 2.33640830216757307397868702436, 3.06636865285086690581996655294, 5.28238609246650201404048796435, 6.69666395743543387323480009985, 7.39711492944729808375769401370, 8.230604512259396147907638537517, 9.327231920788095048915703517626, 9.870087361423128029626641447080, 11.11733554788312746484596141422, 11.87992234514180829312089293244

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