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Properties

Label 2-280-280.117-c1-0-22
Degree $2$
Conductor $280$
Sign $-0.0515 + 0.998i$
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  + (−0.464 − 1.33i)2-s + (−0.0539 − 0.0144i)3-s + (−1.56 + 1.24i)4-s + (1.91 + 1.15i)5-s + (0.00574 + 0.0787i)6-s + (−1.56 − 2.13i)7-s + (2.38 + 1.51i)8-s + (−2.59 − 1.49i)9-s + (0.651 − 3.09i)10-s + (3.24 − 1.87i)11-s + (0.102 − 0.0442i)12-s + (4.30 − 4.30i)13-s + (−2.12 + 3.07i)14-s + (−0.0866 − 0.0898i)15-s + (0.921 − 3.89i)16-s + (5.23 + 1.40i)17-s + ⋯
L(s)  = 1  + (−0.328 − 0.944i)2-s + (−0.0311 − 0.00834i)3-s + (−0.784 + 0.620i)4-s + (0.856 + 0.515i)5-s + (0.00234 + 0.0321i)6-s + (−0.589 − 0.807i)7-s + (0.843 + 0.537i)8-s + (−0.865 − 0.499i)9-s + (0.205 − 0.978i)10-s + (0.978 − 0.564i)11-s + (0.0296 − 0.0127i)12-s + (1.19 − 1.19i)13-s + (−0.569 + 0.822i)14-s + (−0.0223 − 0.0232i)15-s + (0.230 − 0.973i)16-s + (1.26 + 0.339i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0515 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0515 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.0515 + 0.998i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.0515 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.753047 - 0.792895i\)
\(L(\frac12)\) \(\approx\) \(0.753047 - 0.792895i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.464 + 1.33i)T \)
5 \( 1 + (-1.91 - 1.15i)T \)
7 \( 1 + (1.56 + 2.13i)T \)
good3 \( 1 + (0.0539 + 0.0144i)T + (2.59 + 1.5i)T^{2} \)
11 \( 1 + (-3.24 + 1.87i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.30 + 4.30i)T - 13iT^{2} \)
17 \( 1 + (-5.23 - 1.40i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.52 + 0.880i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.28 + 4.81i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 4.27T + 29T^{2} \)
31 \( 1 + (5.03 - 2.90i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.57 - 5.87i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 4.10iT - 41T^{2} \)
43 \( 1 + (-6.73 - 6.73i)T + 43iT^{2} \)
47 \( 1 + (2.06 + 7.69i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (0.656 - 2.44i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-1.62 + 0.936i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.75 + 4.76i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.02 - 1.88i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 4.52T + 71T^{2} \)
73 \( 1 + (-1.26 + 4.72i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (6.17 + 3.56i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.07 - 3.07i)T - 83iT^{2} \)
89 \( 1 + (1.41 - 2.45i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.69 - 7.69i)T - 97iT^{2} \)
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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

−11.33122342147096917350794631757, −10.68747007604363506295273602693, −9.905775174508014047078783062235, −8.989501994109913065054329229812, −8.044534075957248484740762105396, −6.52839637426099194096851515806, −5.66654745544531601804381283079, −3.70840630819106082253047843230, −3.06010335213385806861262443361, −1.07213179938384278645797794052, 1.77282934855843080016570372790, 3.97286644146873331216284713076, 5.60883335220738112208904393728, 5.88443827275580978250261999043, 7.11221748768441627818412729269, 8.467494374193170211209254580470, 9.239654796270630553041259029387, 9.645061599316943104131638508333, 11.08943533168951150983648942737, 12.21711595730851795208886093660

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