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Properties

Label 2-360-9.4-c1-0-10
Degree $2$
Conductor $360$
Sign $-0.0938 + 0.995i$
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  + (1.05 − 1.37i)3-s + (−0.5 + 0.866i)5-s + (−1.51 − 2.63i)7-s + (−0.756 − 2.90i)9-s + (−2.63 − 4.56i)11-s + (0.256 − 0.444i)13-s + (0.657 + 1.60i)15-s + 2.80·17-s + 8.29·19-s + (−5.21 − 0.704i)21-s + (−2.51 + 4.36i)23-s + (−0.499 − 0.866i)25-s + (−4.78 − 2.03i)27-s + (2.39 + 4.14i)29-s + (4.29 − 7.43i)31-s + ⋯
L(s)  = 1  + (0.611 − 0.791i)3-s + (−0.223 + 0.387i)5-s + (−0.573 − 0.994i)7-s + (−0.252 − 0.967i)9-s + (−0.794 − 1.37i)11-s + (0.0712 − 0.123i)13-s + (0.169 + 0.413i)15-s + 0.679·17-s + 1.90·19-s + (−1.13 − 0.153i)21-s + (−0.525 + 0.909i)23-s + (−0.0999 − 0.173i)25-s + (−0.919 − 0.392i)27-s + (0.444 + 0.769i)29-s + (0.771 − 1.33i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0938 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0938 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.0938 + 0.995i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ -0.0938 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.923557 - 1.01473i\)
\(L(\frac12)\) \(\approx\) \(0.923557 - 1.01473i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.05 + 1.37i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
good7 \( 1 + (1.51 + 2.63i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.63 + 4.56i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.256 + 0.444i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 2.80T + 17T^{2} \)
19 \( 1 - 8.29T + 19T^{2} \)
23 \( 1 + (2.51 - 4.36i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.39 - 4.14i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.29 + 7.43i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 6.58T + 37T^{2} \)
41 \( 1 + (3.99 - 6.91i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.598 - 1.03i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.81 - 8.33i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 0.467T + 53T^{2} \)
59 \( 1 + (-0.378 + 0.655i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.135 - 0.234i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.63 + 6.28i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.48T + 71T^{2} \)
73 \( 1 - 5.31T + 73T^{2} \)
79 \( 1 + (-7.98 - 13.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.03 + 10.4i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 15.2T + 89T^{2} \)
97 \( 1 + (-3.18 - 5.51i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.28423101565251463864882084538, −10.22263586717136667663277738813, −9.384036773906066650234320999167, −8.019618300757015665208449497706, −7.62548050314766624438166536650, −6.56332776691329741318823737905, −5.52068989671869698482627987930, −3.57315017993045927528310890337, −3.01852286792293625595308792944, −0.917113918918906231750593021229, 2.31826489946620429477503661461, 3.43706946338846378190013443254, 4.81751247101950162962149712641, 5.50895181848774254341279153161, 7.12870456935367957076688410973, 8.137429137184670835154875998874, 8.993601619420196824563975323800, 9.869834451349746833333962119434, 10.37110859486709705693821922631, 12.00680551961464104190809019503

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