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Properties

Label 2-360-45.4-c1-0-2
Degree $2$
Conductor $360$
Sign $-0.632 - 0.774i$
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  + (1.11 + 1.32i)3-s + (−1.73 + 1.41i)5-s + (−2.19 + 1.26i)7-s + (−0.516 + 2.95i)9-s + (0.162 + 0.280i)11-s + (−4.05 − 2.33i)13-s + (−3.80 − 0.727i)15-s + 5.64i·17-s + 5.56·19-s + (−4.12 − 1.49i)21-s + (−1.27 − 0.736i)23-s + (1.01 − 4.89i)25-s + (−4.49 + 2.60i)27-s + (4.86 + 8.42i)29-s + (3.52 − 6.10i)31-s + ⋯
L(s)  = 1  + (0.643 + 0.765i)3-s + (−0.775 + 0.631i)5-s + (−0.830 + 0.479i)7-s + (−0.172 + 0.985i)9-s + (0.0488 + 0.0846i)11-s + (−1.12 − 0.648i)13-s + (−0.982 − 0.187i)15-s + 1.36i·17-s + 1.27·19-s + (−0.900 − 0.327i)21-s + (−0.265 − 0.153i)23-s + (0.203 − 0.979i)25-s + (−0.864 + 0.501i)27-s + (0.902 + 1.56i)29-s + (0.633 − 1.09i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.632 - 0.774i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.632 - 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.632 - 0.774i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ -0.632 - 0.774i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.470405 + 0.991111i\)
\(L(\frac12)\) \(\approx\) \(0.470405 + 0.991111i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.11 - 1.32i)T \)
5 \( 1 + (1.73 - 1.41i)T \)
good7 \( 1 + (2.19 - 1.26i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.162 - 0.280i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.05 + 2.33i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 5.64iT - 17T^{2} \)
19 \( 1 - 5.56T + 19T^{2} \)
23 \( 1 + (1.27 + 0.736i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.86 - 8.42i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.52 + 6.10i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.20iT - 37T^{2} \)
41 \( 1 + (0.881 - 1.52i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.77 + 1.02i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.98 + 3.45i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 4.99iT - 53T^{2} \)
59 \( 1 + (3.40 - 5.89i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.14 - 8.90i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.27 - 2.46i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.24T + 71T^{2} \)
73 \( 1 + 12.7iT - 73T^{2} \)
79 \( 1 + (8.27 + 14.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-10.9 + 6.30i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 3.66T + 89T^{2} \)
97 \( 1 + (-1.69 + 0.980i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.80936452727477009194577336294, −10.52202252083683448432212321388, −10.08315052760804729946525247470, −9.049352463632876222770761120537, −8.065044579447843238906134210621, −7.24442339227509698531809279348, −5.93369178590227946070120231254, −4.64561177768136595586629697989, −3.44278401213260178104308955972, −2.69264832533135453074789092699, 0.69424948237425071245964087038, 2.67023413132152807556928052806, 3.82222117731274343722431805684, 5.07771150502224386833176506284, 6.68602132142591307388580159433, 7.34598613929974633482557107227, 8.152975723163514357426772588868, 9.353369270520629937628169229578, 9.780638328409302500947307254128, 11.52582319446972759496354774783

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