↑

Properties

Label 2-280-40.29-c1-0-0
Degree $2$
Conductor $280$
Sign $0.211 - 0.977i$
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  + (0.269 − 1.38i)2-s − 2.55·3-s + (−1.85 − 0.747i)4-s + (−0.790 − 2.09i)5-s + (−0.687 + 3.54i)6-s + i·7-s + (−1.53 + 2.37i)8-s + 3.53·9-s + (−3.11 + 0.535i)10-s + 2.94i·11-s + (4.74 + 1.90i)12-s − 1.88·13-s + (1.38 + 0.269i)14-s + (2.02 + 5.34i)15-s + (2.88 + 2.77i)16-s − 1.25i·17-s + ⋯
L(s)  = 1  + (0.190 − 0.981i)2-s − 1.47·3-s + (−0.927 − 0.373i)4-s + (−0.353 − 0.935i)5-s + (−0.280 + 1.44i)6-s + 0.377i·7-s + (−0.543 + 0.839i)8-s + 1.17·9-s + (−0.985 + 0.169i)10-s + 0.886i·11-s + (1.36 + 0.551i)12-s − 0.522·13-s + (0.371 + 0.0719i)14-s + (0.521 + 1.38i)15-s + (0.720 + 0.693i)16-s − 0.304i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.211 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.211 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.211 - 0.977i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.211 - 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0775688 + 0.0625973i\)
\(L(\frac12)\) \(\approx\) \(0.0775688 + 0.0625973i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.269 + 1.38i)T \)
5 \( 1 + (0.790 + 2.09i)T \)
7 \( 1 - iT \)
good3 \( 1 + 2.55T + 3T^{2} \)
11 \( 1 - 2.94iT - 11T^{2} \)
13 \( 1 + 1.88T + 13T^{2} \)
17 \( 1 + 1.25iT - 17T^{2} \)
19 \( 1 - 2.48iT - 19T^{2} \)
23 \( 1 - 2.92iT - 23T^{2} \)
29 \( 1 - 0.808iT - 29T^{2} \)
31 \( 1 + 10.5T + 31T^{2} \)
37 \( 1 + 2.08T + 37T^{2} \)
41 \( 1 + 9.10T + 41T^{2} \)
43 \( 1 + 11.5T + 43T^{2} \)
47 \( 1 + 10.2iT - 47T^{2} \)
53 \( 1 - 9.17T + 53T^{2} \)
59 \( 1 - 10.0iT - 59T^{2} \)
61 \( 1 + 10.3iT - 61T^{2} \)
67 \( 1 - 2.89T + 67T^{2} \)
71 \( 1 - 11.4T + 71T^{2} \)
73 \( 1 - 16.4iT - 73T^{2} \)
79 \( 1 + 13.0T + 79T^{2} \)
83 \( 1 + 8.79T + 83T^{2} \)
89 \( 1 - 6.64T + 89T^{2} \)
97 \( 1 - 7.33iT - 97T^{2} \)
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

−11.93906902955670796802694955791, −11.49735513322273601749319074369, −10.32911136247430424791789789815, −9.584534045830895088431390102052, −8.473099329562841386277143719982, −7.02699512605768902674246938735, −5.39539849553732837262633789641, −5.14165294459786093975917335091, −3.86979894805102563886735359876, −1.70155969271568254816544564673, 0.085625206302210314340274316935, 3.48051365189747284177197920928, 4.77852842802864497572653374892, 5.78898609608008629119187965061, 6.64419455260691418345505093964, 7.30524943780899686076308761230, 8.523393276348992677597801276448, 9.970823939821059776297336617805, 10.83545799597643291533714273022, 11.59080286202996367580311327111

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