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Properties

Label 2-360-72.11-c1-0-14
Degree $2$
Conductor $360$
Sign $0.630 - 0.776i$
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  + (−1.40 + 0.139i)2-s + (1.36 + 1.06i)3-s + (1.96 − 0.393i)4-s + (−0.5 − 0.866i)5-s + (−2.06 − 1.31i)6-s + (1.02 + 0.589i)7-s + (−2.70 + 0.827i)8-s + (0.720 + 2.91i)9-s + (0.824 + 1.14i)10-s + (1.46 + 0.846i)11-s + (3.09 + 1.55i)12-s + (0.622 − 0.359i)13-s + (−1.52 − 0.687i)14-s + (0.242 − 1.71i)15-s + (3.69 − 1.54i)16-s + 4.05i·17-s + ⋯
L(s)  = 1  + (−0.995 + 0.0988i)2-s + (0.787 + 0.616i)3-s + (0.980 − 0.196i)4-s + (−0.223 − 0.387i)5-s + (−0.844 − 0.535i)6-s + (0.386 + 0.222i)7-s + (−0.956 + 0.292i)8-s + (0.240 + 0.970i)9-s + (0.260 + 0.363i)10-s + (0.442 + 0.255i)11-s + (0.893 + 0.449i)12-s + (0.172 − 0.0996i)13-s + (−0.406 − 0.183i)14-s + (0.0626 − 0.442i)15-s + (0.922 − 0.385i)16-s + 0.983i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.630 - 0.776i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.630 - 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.630 - 0.776i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ 0.630 - 0.776i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06030 + 0.504726i\)
\(L(\frac12)\) \(\approx\) \(1.06030 + 0.504726i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 - 0.139i)T \)
3 \( 1 + (-1.36 - 1.06i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
good7 \( 1 + (-1.02 - 0.589i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.46 - 0.846i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.622 + 0.359i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 4.05iT - 17T^{2} \)
19 \( 1 - 5.64T + 19T^{2} \)
23 \( 1 + (2.06 + 3.57i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.77 + 3.07i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.580 - 0.334i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 11.3iT - 37T^{2} \)
41 \( 1 + (-0.943 + 0.544i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.39 + 2.40i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.66 - 4.61i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 9.15T + 53T^{2} \)
59 \( 1 + (-6.11 + 3.53i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.33 - 1.34i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.74 + 13.4i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.1T + 71T^{2} \)
73 \( 1 + 7.59T + 73T^{2} \)
79 \( 1 + (11.3 + 6.53i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (12.0 + 6.93i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 4.96iT - 89T^{2} \)
97 \( 1 + (0.862 - 1.49i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.40330427824769661601540002242, −10.30335152069206751864643857023, −9.702403858847580912567662144649, −8.649378704327023165499430225972, −8.222381069940179846240829948241, −7.20508256045248459018089726787, −5.83256158140619132984594769645, −4.50257321159289308856031586482, −3.13097530630389213779971011151, −1.65635702708297677035324402966, 1.19098137938308988933280055541, 2.66030449199135989309420572013, 3.74817180149124954684882255483, 5.81211269101521101968596013297, 7.17915295362147528935909025270, 7.42650631845626867838060211755, 8.573151390856150620524893207112, 9.306874470578642627802517934536, 10.19256543827293040317485383003, 11.47057515663458761261984332184

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