↑

Properties

Label 2-280-280.107-c1-0-10
Degree $2$
Conductor $280$
Sign $0.793 - 0.608i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.41 + 0.0614i)2-s + (−0.862 − 0.231i)3-s + (1.99 − 0.173i)4-s + (1.04 + 1.97i)5-s + (1.23 + 0.273i)6-s + (2.36 − 1.19i)7-s + (−2.80 + 0.367i)8-s + (−1.90 − 1.10i)9-s + (−1.59 − 2.72i)10-s + (2.58 + 4.47i)11-s + (−1.75 − 0.310i)12-s + (−2.64 − 2.64i)13-s + (−3.26 + 1.83i)14-s + (−0.443 − 1.94i)15-s + (3.93 − 0.691i)16-s + (−0.725 + 2.70i)17-s + ⋯
L(s)  = 1  + (−0.999 + 0.0434i)2-s + (−0.497 − 0.133i)3-s + (0.996 − 0.0868i)4-s + (0.466 + 0.884i)5-s + (0.503 + 0.111i)6-s + (0.892 − 0.450i)7-s + (−0.991 + 0.129i)8-s + (−0.636 − 0.367i)9-s + (−0.504 − 0.863i)10-s + (0.779 + 1.35i)11-s + (−0.507 − 0.0896i)12-s + (−0.732 − 0.732i)13-s + (−0.872 + 0.489i)14-s + (−0.114 − 0.502i)15-s + (0.984 − 0.172i)16-s + (−0.175 + 0.656i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 - 0.608i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.793 - 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.793 - 0.608i$
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: \((2,\ 280,\ (\ :1/2),\ 0.793 - 0.608i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.773775 + 0.262281i\)
\(L(\frac12)\) \(\approx\) \(0.773775 + 0.262281i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 - 0.0614i)T \)
5 \( 1 + (-1.04 - 1.97i)T \)
7 \( 1 + (-2.36 + 1.19i)T \)
good3 \( 1 + (0.862 + 0.231i)T + (2.59 + 1.5i)T^{2} \)
11 \( 1 + (-2.58 - 4.47i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.64 + 2.64i)T + 13iT^{2} \)
17 \( 1 + (0.725 - 2.70i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-5.82 - 3.36i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.16 + 1.38i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 8.59T + 29T^{2} \)
31 \( 1 + (1.82 - 1.05i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.450 - 1.67i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 2.04T + 41T^{2} \)
43 \( 1 + (3.38 - 3.38i)T - 43iT^{2} \)
47 \( 1 + (0.488 + 1.82i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (0.759 - 2.83i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (5.92 - 3.41i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.25 + 3.61i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.41 + 8.99i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 11.8iT - 71T^{2} \)
73 \( 1 + (-2.45 - 0.657i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-5.08 + 8.81i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.87 - 1.87i)T - 83iT^{2} \)
89 \( 1 + (12.9 + 7.48i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.762 - 0.762i)T + 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.78778985368266241790263297051, −10.81408950874721606119482837471, −10.16713132032382528116576275405, −9.277143684472987311451935757467, −7.969524998839674960842786243210, −7.11940614962843425570595391719, −6.32485045062210595060214767863, −5.07314803775346028133223837134, −3.06434862576505801140844814893, −1.50525935180132593980941304721, 1.05013013506495288199228635831, 2.70736384219705288869342118482, 4.92152364259584074230197640658, 5.69216369663764079460479828244, 6.92232815459355217927174958599, 8.273158407971651653509575650398, 8.907454715225341004587071348907, 9.618504397790478501223704957057, 11.02276086424127249452158342948, 11.57448046142695011655333260220

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