Properties

Label 2-360-72.59-c1-0-10
Degree $2$
Conductor $360$
Sign $0.712 - 0.701i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.33 + 0.464i)2-s + (−1.29 − 1.14i)3-s + (1.56 − 1.23i)4-s + (0.5 − 0.866i)5-s + (2.26 + 0.929i)6-s + (−3.14 + 1.81i)7-s + (−1.52 + 2.38i)8-s + (0.368 + 2.97i)9-s + (−0.266 + 1.38i)10-s + (−2.02 + 1.17i)11-s + (−3.45 − 0.190i)12-s + (4.70 + 2.71i)13-s + (3.35 − 3.87i)14-s + (−1.64 + 0.550i)15-s + (0.925 − 3.89i)16-s − 4.45i·17-s + ⋯
L(s)  = 1  + (−0.944 + 0.328i)2-s + (−0.749 − 0.662i)3-s + (0.784 − 0.619i)4-s + (0.223 − 0.387i)5-s + (0.925 + 0.379i)6-s + (−1.18 + 0.685i)7-s + (−0.537 + 0.843i)8-s + (0.122 + 0.992i)9-s + (−0.0841 + 0.439i)10-s + (−0.611 + 0.353i)11-s + (−0.998 − 0.0550i)12-s + (1.30 + 0.753i)13-s + (0.896 − 1.03i)14-s + (−0.424 + 0.142i)15-s + (0.231 − 0.972i)16-s − 1.07i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.712 - 0.701i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.712 - 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.712 - 0.701i$
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: \((2,\ 360,\ (\ :1/2),\ 0.712 - 0.701i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.530819 + 0.217332i\)
\(L(\frac12)\) \(\approx\) \(0.530819 + 0.217332i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.33 - 0.464i)T \)
3 \( 1 + (1.29 + 1.14i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
good7 \( 1 + (3.14 - 1.81i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.02 - 1.17i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.70 - 2.71i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 4.45iT - 17T^{2} \)
19 \( 1 - 7.29T + 19T^{2} \)
23 \( 1 + (2.97 - 5.15i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.52 - 2.63i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-6.43 - 3.71i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.85iT - 37T^{2} \)
41 \( 1 + (-0.201 - 0.116i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.490 + 0.850i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.48 - 2.57i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 11.3T + 53T^{2} \)
59 \( 1 + (-1.02 - 0.590i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (10.1 - 5.87i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.00 - 3.47i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.10T + 71T^{2} \)
73 \( 1 + 1.90T + 73T^{2} \)
79 \( 1 + (0.784 - 0.452i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-11.9 + 6.92i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 12.5iT - 89T^{2} \)
97 \( 1 + (-6.51 - 11.2i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.82015376746035688136917566778, −10.47025644951339922768280121206, −9.626516445637054333750329624849, −8.869492302929587931616774937022, −7.69987453831952203844095695503, −6.80125068250328262772161403684, −5.97070522851720237585567081815, −5.18588328098062529566924911894, −2.83823468241838565358294544399, −1.24584685293935797084878325766, 0.67537566246916492348650032302, 3.06587528679117491781476960054, 3.88622742637872580120174165200, 5.89149220476753759730489984696, 6.39754674215484464510980728635, 7.63881285726595875331991914081, 8.748609721554413800673751443909, 9.917034466609161411016925875302, 10.30274698896941939268400898026, 10.93408248953674823195712054655

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