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Properties

Label 2-360-360.349-c1-0-48
Degree $2$
Conductor $360$
Sign $-0.967 - 0.254i$
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  + (−0.954 − 1.04i)2-s + (−1.07 − 1.36i)3-s + (−0.179 + 1.99i)4-s + (1.96 + 1.06i)5-s + (−0.400 + 2.41i)6-s + (−1.60 − 0.926i)7-s + (2.25 − 1.71i)8-s + (−0.708 + 2.91i)9-s + (−0.770 − 3.06i)10-s + (−2.88 − 1.66i)11-s + (2.90 − 1.88i)12-s + (−3.23 − 5.59i)13-s + (0.563 + 2.55i)14-s + (−0.661 − 3.81i)15-s + (−3.93 − 0.715i)16-s + 0.590i·17-s + ⋯
L(s)  = 1  + (−0.674 − 0.738i)2-s + (−0.617 − 0.786i)3-s + (−0.0897 + 0.995i)4-s + (0.880 + 0.474i)5-s + (−0.163 + 0.986i)6-s + (−0.606 − 0.350i)7-s + (0.795 − 0.605i)8-s + (−0.236 + 0.971i)9-s + (−0.243 − 0.969i)10-s + (−0.870 − 0.502i)11-s + (0.838 − 0.544i)12-s + (−0.896 − 1.55i)13-s + (0.150 + 0.683i)14-s + (−0.170 − 0.985i)15-s + (−0.983 − 0.178i)16-s + 0.143i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 - 0.254i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.967 - 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.967 - 0.254i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ -0.967 - 0.254i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0498178 + 0.385678i\)
\(L(\frac12)\) \(\approx\) \(0.0498178 + 0.385678i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.954 + 1.04i)T \)
3 \( 1 + (1.07 + 1.36i)T \)
5 \( 1 + (-1.96 - 1.06i)T \)
good7 \( 1 + (1.60 + 0.926i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.88 + 1.66i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.23 + 5.59i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 0.590iT - 17T^{2} \)
19 \( 1 + 4.73iT - 19T^{2} \)
23 \( 1 + (7.05 - 4.07i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.135 - 0.0783i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.77 + 4.80i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 3.21T + 37T^{2} \)
41 \( 1 + (0.929 + 1.60i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.08 - 1.88i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-8.32 - 4.80i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 6.40T + 53T^{2} \)
59 \( 1 + (-1.67 + 0.967i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.78 - 1.60i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.33 + 5.77i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.20T + 71T^{2} \)
73 \( 1 + 15.2iT - 73T^{2} \)
79 \( 1 + (0.603 - 1.04i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.12 + 3.67i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 12.3T + 89T^{2} \)
97 \( 1 + (-12.0 - 6.95i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.68528008978861409118538740130, −10.32880210327902081398028913999, −9.379769612660714968174838940152, −7.944378429061801579611489020228, −7.37325885882283734563173854953, −6.19091890293297631347248347320, −5.18381887178033394309744515558, −3.16304191332547646052540518396, −2.19061163265527265369894990718, −0.33015001950027950405708526056, 2.08970275802706773818454414236, 4.38869907148160062151540747076, 5.30772422781567137888358281867, 6.13116364742178742062101147288, 7.00389520718954612360386165058, 8.466388895701579509478566337353, 9.365332104943488394610499545324, 9.946114403291409240340426800644, 10.49147067325181024808310853159, 11.91548834148916984640151449444

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