↑

Properties

Label 2-360-72.59-c1-0-4
Degree $2$
Conductor $360$
Sign $-0.680 + 0.732i$
Analytic cond. $2.87461$
Root an. cond. $1.69546$
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  + (0.652 + 1.25i)2-s + (−1.71 − 0.230i)3-s + (−1.14 + 1.63i)4-s + (−0.5 + 0.866i)5-s + (−0.831 − 2.30i)6-s + (−1.06 + 0.616i)7-s + (−2.80 − 0.374i)8-s + (2.89 + 0.789i)9-s + (−1.41 − 0.0625i)10-s + (−0.850 + 0.490i)11-s + (2.34 − 2.54i)12-s + (−3.48 − 2.01i)13-s + (−1.46 − 0.937i)14-s + (1.05 − 1.37i)15-s + (−1.35 − 3.76i)16-s − 6.50i·17-s + ⋯
L(s)  = 1  + (0.461 + 0.887i)2-s + (−0.991 − 0.132i)3-s + (−0.574 + 0.818i)4-s + (−0.223 + 0.387i)5-s + (−0.339 − 0.940i)6-s + (−0.403 + 0.232i)7-s + (−0.991 − 0.132i)8-s + (0.964 + 0.263i)9-s + (−0.446 − 0.0197i)10-s + (−0.256 + 0.147i)11-s + (0.678 − 0.734i)12-s + (−0.966 − 0.558i)13-s + (−0.392 − 0.250i)14-s + (0.273 − 0.354i)15-s + (−0.339 − 0.940i)16-s − 1.57i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.680 + 0.732i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.680 + 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.680 + 0.732i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ -0.680 + 0.732i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.113936 - 0.261230i\)
\(L(\frac12)\) \(\approx\) \(0.113936 - 0.261230i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.652 - 1.25i)T \)
3 \( 1 + (1.71 + 0.230i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
good7 \( 1 + (1.06 - 0.616i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.850 - 0.490i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.48 + 2.01i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 6.50iT - 17T^{2} \)
19 \( 1 + 5.33T + 19T^{2} \)
23 \( 1 + (3.93 - 6.80i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.07 - 5.32i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.79 + 1.03i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.78iT - 37T^{2} \)
41 \( 1 + (-9.01 - 5.20i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.25 + 3.90i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.53 - 2.65i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 2.52T + 53T^{2} \)
59 \( 1 + (-2.31 - 1.33i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (11.9 - 6.92i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.368 + 0.638i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.48T + 71T^{2} \)
73 \( 1 + 3.14T + 73T^{2} \)
79 \( 1 + (5.93 - 3.42i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.85 - 1.65i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 15.1iT - 89T^{2} \)
97 \( 1 + (-5.02 - 8.70i)T + (-48.5 + 84.0i)T^{2} \)
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

−12.17133929963928840488572321986, −11.35080278224012176412956436093, −10.15177890787526441488198942463, −9.289058338648803519807563249388, −7.79262743006707451405869256235, −7.18442892310749048810256481153, −6.24079402836545412625648044613, −5.31626011351064177907817614196, −4.41941045736553225803851729287, −2.88058172431668642377059771040, 0.17990100391168719113005951090, 2.08853592558092269352439126556, 4.00049270253702632616014123218, 4.56755018657433275982144499779, 5.85692440750847139372081775388, 6.59950804616837614034544620348, 8.195693885451603211907149130478, 9.384080693985114559146741966043, 10.35176050386604127647617004803, 10.78563451092342056223345240539

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