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Properties

Label 2-360-120.77-c1-0-20
Degree $2$
Conductor $360$
Sign $-0.897 + 0.440i$
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  + (−1.16 + 0.803i)2-s + (0.709 − 1.87i)4-s + (−1.29 − 1.82i)5-s + (−0.306 − 0.306i)7-s + (0.676 + 2.74i)8-s + (2.97 + 1.08i)10-s − 4.06·11-s + (0.625 + 0.625i)13-s + (0.603 + 0.110i)14-s + (−2.99 − 2.65i)16-s + (−3.57 + 3.57i)17-s − 6.82·19-s + (−4.32 + 1.12i)20-s + (4.73 − 3.26i)22-s + (1.58 + 1.58i)23-s + ⋯
L(s)  = 1  + (−0.822 + 0.568i)2-s + (0.354 − 0.935i)4-s + (−0.578 − 0.815i)5-s + (−0.115 − 0.115i)7-s + (0.239 + 0.970i)8-s + (0.939 + 0.343i)10-s − 1.22·11-s + (0.173 + 0.173i)13-s + (0.161 + 0.0295i)14-s + (−0.748 − 0.663i)16-s + (−0.867 + 0.867i)17-s − 1.56·19-s + (−0.967 + 0.251i)20-s + (1.00 − 0.696i)22-s + (0.331 + 0.331i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.897 + 0.440i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.897 + 0.440i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.897 + 0.440i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ -0.897 + 0.440i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0260484 - 0.112197i\)
\(L(\frac12)\) \(\approx\) \(0.0260484 - 0.112197i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.16 - 0.803i)T \)
3 \( 1 \)
5 \( 1 + (1.29 + 1.82i)T \)
good7 \( 1 + (0.306 + 0.306i)T + 7iT^{2} \)
11 \( 1 + 4.06T + 11T^{2} \)
13 \( 1 + (-0.625 - 0.625i)T + 13iT^{2} \)
17 \( 1 + (3.57 - 3.57i)T - 17iT^{2} \)
19 \( 1 + 6.82T + 19T^{2} \)
23 \( 1 + (-1.58 - 1.58i)T + 23iT^{2} \)
29 \( 1 + 8.50iT - 29T^{2} \)
31 \( 1 + 2.56T + 31T^{2} \)
37 \( 1 + (1.69 - 1.69i)T - 37iT^{2} \)
41 \( 1 + 5.19iT - 41T^{2} \)
43 \( 1 + (4.18 + 4.18i)T + 43iT^{2} \)
47 \( 1 + (4.58 - 4.58i)T - 47iT^{2} \)
53 \( 1 + (-7.41 + 7.41i)T - 53iT^{2} \)
59 \( 1 - 8.79iT - 59T^{2} \)
61 \( 1 - 6.08iT - 61T^{2} \)
67 \( 1 + (6.18 - 6.18i)T - 67iT^{2} \)
71 \( 1 + 14.7iT - 71T^{2} \)
73 \( 1 + (-3.05 + 3.05i)T - 73iT^{2} \)
79 \( 1 - 8.56iT - 79T^{2} \)
83 \( 1 + (5.13 - 5.13i)T - 83iT^{2} \)
89 \( 1 - 9.88T + 89T^{2} \)
97 \( 1 + (-1.75 - 1.75i)T + 97iT^{2} \)
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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

−10.82361302960639539925211938576, −10.11592521465099039947416501128, −8.860086119640267706373463508236, −8.365823692798070736793416923172, −7.47897618450715762832676372001, −6.35155603398336025222615810242, −5.26658914497691584884091810053, −4.15455978673994408514477266812, −2.07565821742779477226971402110, −0.094769135482843300640491593713, 2.34499247116244638269536401982, 3.29222295759691890355538555104, 4.68632455365539535722668970217, 6.46340157005826478482149637969, 7.28397725412759720597087322217, 8.213677944566050634633229611685, 9.026175511549801623293075903442, 10.29316746172116581281261298359, 10.80526605431707978832254884233, 11.48440418402234346377165710592

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