Properties

Label 2-360-45.4-c1-0-12
Degree $2$
Conductor $360$
Sign $0.454 + 0.890i$
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  + (1.27 − 1.16i)3-s + (−2.07 + 0.840i)5-s + (3.28 − 1.89i)7-s + (0.264 − 2.98i)9-s + (−1.86 − 3.23i)11-s + (1.09 + 0.634i)13-s + (−1.66 + 3.49i)15-s − 0.973i·17-s + 2.74·19-s + (1.97 − 6.26i)21-s + (5.83 + 3.36i)23-s + (3.58 − 3.48i)25-s + (−3.15 − 4.12i)27-s + (−1.99 − 3.45i)29-s + (−5.36 + 9.28i)31-s + ⋯
L(s)  = 1  + (0.737 − 0.675i)3-s + (−0.926 + 0.375i)5-s + (1.24 − 0.716i)7-s + (0.0882 − 0.996i)9-s + (−0.562 − 0.974i)11-s + (0.304 + 0.176i)13-s + (−0.429 + 0.902i)15-s − 0.236i·17-s + 0.629·19-s + (0.431 − 1.36i)21-s + (1.21 + 0.702i)23-s + (0.717 − 0.696i)25-s + (−0.607 − 0.794i)27-s + (−0.370 − 0.641i)29-s + (−0.962 + 1.66i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.454 + 0.890i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.454 + 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.454 + 0.890i$
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: \((2,\ 360,\ (\ :1/2),\ 0.454 + 0.890i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36673 - 0.837225i\)
\(L(\frac12)\) \(\approx\) \(1.36673 - 0.837225i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.27 + 1.16i)T \)
5 \( 1 + (2.07 - 0.840i)T \)
good7 \( 1 + (-3.28 + 1.89i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.86 + 3.23i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.09 - 0.634i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 0.973iT - 17T^{2} \)
19 \( 1 - 2.74T + 19T^{2} \)
23 \( 1 + (-5.83 - 3.36i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.99 + 3.45i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (5.36 - 9.28i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 7.59iT - 37T^{2} \)
41 \( 1 + (2.92 - 5.07i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.95 - 2.86i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.18 + 2.41i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 10.1iT - 53T^{2} \)
59 \( 1 + (3.98 - 6.91i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.29 - 7.44i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.993 + 0.573i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.20T + 71T^{2} \)
73 \( 1 - 0.990iT - 73T^{2} \)
79 \( 1 + (-7.83 - 13.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.08 - 0.625i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 8.97T + 89T^{2} \)
97 \( 1 + (-5.56 + 3.21i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.23933572745250448236659887966, −10.70182619865454276409882132695, −9.127552390405448931069295233804, −8.258291011192870636511464967254, −7.58877901125249527775362649665, −6.93098786737945455679802997985, −5.33490689304147271098150980318, −3.95624566877249393364321645991, −2.97318882738867386672803053205, −1.17697316473679204308597369194, 2.02211543497790164527036727395, 3.47777130256337791108794421852, 4.73278716803567409097340877267, 5.22033927525894805198030519183, 7.26437688355232340619386580934, 8.068299674011016011555457291455, 8.673737993124648577275681829237, 9.617855893628769107552312897231, 10.81155015291015364333806697021, 11.43830020985548921713527003587

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