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Properties

Label 2-360-45.4-c1-0-0
Degree $2$
Conductor $360$
Sign $-0.974 + 0.222i$
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 + (0.307 + 2.21i)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 + (−2.98 − 2.46i)15-s + 0.973i·17-s + 2.74·19-s + (1.97 − 6.26i)21-s + (−5.83 − 3.36i)23-s + (−4.81 + 1.36i)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 + (0.137 + 0.990i)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 + (−0.770 − 0.637i)15-s + 0.236i·17-s + 0.629·19-s + (0.431 − 1.36i)21-s + (−1.21 − 0.702i)23-s + (−0.962 + 0.272i)25-s + (0.607 + 0.794i)27-s + (−0.370 − 0.641i)29-s + (−0.962 + 1.66i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 + 0.222i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.974 + 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.974 + 0.222i$
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: \((2,\ 360,\ (\ :1/2),\ -0.974 + 0.222i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0379226 - 0.336692i\)
\(L(\frac12)\) \(\approx\) \(0.0379226 - 0.336692i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.307 - 2.21i)T \)
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

−11.81495166108392686812664038389, −10.90192713795466771433135311278, −10.12562192053837299874212645078, −9.533187047172640708974322420852, −8.308581783907912737947042165565, −6.83536535355761550388432337033, −6.10357653968795985912408751401, −5.35265564827187458794087275012, −3.65324616031228845195363012868, −2.81966146481237173331387829853, 0.23748366646979550554632543074, 1.99879160310702784052968958870, 3.94725999473629772376091706467, 5.16277471498769774852212757103, 6.04233163872715351032746981915, 7.23955508524299487346221442438, 7.76801388658556609746813512284, 9.400052578319845176319433725454, 9.893440029707795092546604954043, 11.01684528772255215736771410812

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