Properties

Label 2-360-8.5-c1-0-8
Degree $2$
Conductor $360$
Sign $0.883 + 0.467i$
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  + (−1.32 + 0.5i)2-s + (1.50 − 1.32i)4-s i·5-s + 2·7-s + (−1.32 + 2.50i)8-s + (0.5 + 1.32i)10-s − 2i·11-s + (−2.64 + i)14-s + (0.500 − 3.96i)16-s − 5.29i·19-s + (−1.32 − 1.50i)20-s + (1 + 2.64i)22-s + 5.29·23-s − 25-s + (3.00 − 2.64i)28-s − 6i·29-s + ⋯
L(s)  = 1  + (−0.935 + 0.353i)2-s + (0.750 − 0.661i)4-s − 0.447i·5-s + 0.755·7-s + (−0.467 + 0.883i)8-s + (0.158 + 0.418i)10-s − 0.603i·11-s + (−0.707 + 0.267i)14-s + (0.125 − 0.992i)16-s − 1.21i·19-s + (−0.295 − 0.335i)20-s + (0.213 + 0.564i)22-s + 1.10·23-s − 0.200·25-s + (0.566 − 0.499i)28-s − 1.11i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.883 + 0.467i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.883 + 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.883 + 0.467i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ 0.883 + 0.467i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.918322 - 0.227989i\)
\(L(\frac12)\) \(\approx\) \(0.918322 - 0.227989i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.32 - 0.5i)T \)
3 \( 1 \)
5 \( 1 + iT \)
good7 \( 1 - 2T + 7T^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 5.29iT - 19T^{2} \)
23 \( 1 - 5.29T + 23T^{2} \)
29 \( 1 + 6iT - 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 - 10.5iT - 37T^{2} \)
41 \( 1 - 10.5T + 41T^{2} \)
43 \( 1 + 10.5iT - 43T^{2} \)
47 \( 1 - 5.29T + 47T^{2} \)
53 \( 1 - 2iT - 53T^{2} \)
59 \( 1 + 10iT - 59T^{2} \)
61 \( 1 - 10.5iT - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 10.5T + 71T^{2} \)
73 \( 1 + 14T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 + 10.5T + 89T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−11.27132196395851703726953625619, −10.41337216527818415061976883347, −9.294464979345681713024557015371, −8.609013423775134402833651769980, −7.79978927213786820934880663446, −6.77137566652881648478023732742, −5.63699143439207591158254968529, −4.61668749150245551957992467265, −2.64320459199366440824974384142, −0.994883475537477623186803082558, 1.54740601209356671730314881445, 2.91127859826938712957343690749, 4.32428788257918239087071333258, 5.87982453619531752534086852503, 7.12877778983269025059666269001, 7.78012344970630886962265511538, 8.801656035346736390745595073924, 9.712828963180114982055977011016, 10.65765938299176352109103215428, 11.22199967987096879872741092170

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