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Properties

Label 2-360-8.5-c1-0-14
Degree $2$
Conductor $360$
Sign $0.0864 + 0.996i$
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  + (0.671 − 1.24i)2-s + (−1.09 − 1.67i)4-s + i·5-s + 4.68·7-s + (−2.81 + 0.244i)8-s + (1.24 + 0.671i)10-s − 2.29i·11-s − 4.97i·13-s + (3.14 − 5.83i)14-s + (−1.58 + 3.67i)16-s + 2.97·17-s + 2.68i·19-s + (1.67 − 1.09i)20-s + (−2.85 − 1.53i)22-s − 2.68·23-s + ⋯
L(s)  = 1  + (0.474 − 0.880i)2-s + (−0.549 − 0.835i)4-s + 0.447i·5-s + 1.77·7-s + (−0.996 + 0.0864i)8-s + (0.393 + 0.212i)10-s − 0.691i·11-s − 1.38i·13-s + (0.840 − 1.55i)14-s + (−0.396 + 0.917i)16-s + 0.722·17-s + 0.616i·19-s + (0.373 − 0.245i)20-s + (−0.608 − 0.328i)22-s − 0.560·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0864 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0864 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.0864 + 0.996i$
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: \((2,\ 360,\ (\ :1/2),\ 0.0864 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34199 - 1.23053i\)
\(L(\frac12)\) \(\approx\) \(1.34199 - 1.23053i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.671 + 1.24i)T \)
3 \( 1 \)
5 \( 1 - iT \)
good7 \( 1 - 4.68T + 7T^{2} \)
11 \( 1 + 2.29iT - 11T^{2} \)
13 \( 1 + 4.97iT - 13T^{2} \)
17 \( 1 - 2.97T + 17T^{2} \)
19 \( 1 - 2.68iT - 19T^{2} \)
23 \( 1 + 2.68T + 23T^{2} \)
29 \( 1 + 2iT - 29T^{2} \)
31 \( 1 + 6.97T + 31T^{2} \)
37 \( 1 - 4.39iT - 37T^{2} \)
41 \( 1 - 11.3T + 41T^{2} \)
43 \( 1 - 9.37iT - 43T^{2} \)
47 \( 1 + 7.27T + 47T^{2} \)
53 \( 1 - 2iT - 53T^{2} \)
59 \( 1 + 1.70iT - 59T^{2} \)
61 \( 1 - 4.58iT - 61T^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 + 0.585T + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 - 1.02T + 79T^{2} \)
83 \( 1 - 13.3iT - 83T^{2} \)
89 \( 1 + 3.37T + 89T^{2} \)
97 \( 1 + 3.95T + 97T^{2} \)
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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

−11.15109735860260138593134978401, −10.67342880166814797507092010979, −9.695639661358702937732510933528, −8.326030657087552751849245408891, −7.76124482420859914612933453634, −5.92755438926476484713544937907, −5.25391759228129386195291215817, −4.02794203578242678552985584258, −2.80427027647620278757369386090, −1.33880697152159371453870731140, 1.92238211221686359444567166570, 4.05416027035786545037430141463, 4.78515084122443021577770588885, 5.64019851830828581902514126793, 7.06807342171987581149774966595, 7.74638157401078419942619519805, 8.702442886389727881969116131046, 9.447228835457842445813650071525, 11.00214451820639053698549608874, 11.84048325487958638104946549622

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