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Properties

Label 2-666-111.56-c1-0-2
Degree $2$
Conductor $666$
Sign $-0.982 + 0.187i$
Analytic cond. $5.31803$
Root an. cond. $2.30608$
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  + (0.0871 + 0.996i)2-s + (−0.984 + 0.173i)4-s + (0.0741 − 0.0345i)5-s + (−1.92 − 0.700i)7-s + (−0.258 − 0.965i)8-s + (0.0408 + 0.0708i)10-s + (−0.0467 + 0.0809i)11-s + (−3.00 + 4.29i)13-s + (0.530 − 1.97i)14-s + (0.939 − 0.342i)16-s + (2.94 + 4.21i)17-s + (−7.61 − 0.666i)19-s + (−0.0669 + 0.0468i)20-s + (−0.0847 − 0.0395i)22-s + (−6.72 − 1.80i)23-s + ⋯
L(s)  = 1  + (0.0616 + 0.704i)2-s + (−0.492 + 0.0868i)4-s + (0.0331 − 0.0154i)5-s + (−0.727 − 0.264i)7-s + (−0.0915 − 0.341i)8-s + (0.0129 + 0.0223i)10-s + (−0.0140 + 0.0244i)11-s + (−0.833 + 1.19i)13-s + (0.141 − 0.529i)14-s + (0.234 − 0.0855i)16-s + (0.715 + 1.02i)17-s + (−1.74 − 0.152i)19-s + (−0.0149 + 0.0104i)20-s + (−0.0180 − 0.00842i)22-s + (−1.40 − 0.375i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 + 0.187i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.982 + 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $-0.982 + 0.187i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (611, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ -0.982 + 0.187i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0443399 - 0.468175i\)
\(L(\frac12)\) \(\approx\) \(0.0443399 - 0.468175i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0871 - 0.996i)T \)
3 \( 1 \)
37 \( 1 + (6.08 - 0.146i)T \)
good5 \( 1 + (-0.0741 + 0.0345i)T + (3.21 - 3.83i)T^{2} \)
7 \( 1 + (1.92 + 0.700i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (0.0467 - 0.0809i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.00 - 4.29i)T + (-4.44 - 12.2i)T^{2} \)
17 \( 1 + (-2.94 - 4.21i)T + (-5.81 + 15.9i)T^{2} \)
19 \( 1 + (7.61 + 0.666i)T + (18.7 + 3.29i)T^{2} \)
23 \( 1 + (6.72 + 1.80i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-3.03 + 0.813i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (0.356 - 0.356i)T - 31iT^{2} \)
41 \( 1 + (0.847 + 4.80i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-5.17 - 5.17i)T + 43iT^{2} \)
47 \( 1 + (-7.78 + 4.49i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.03 - 5.57i)T + (-40.6 + 34.0i)T^{2} \)
59 \( 1 + (3.74 - 8.02i)T + (-37.9 - 45.1i)T^{2} \)
61 \( 1 + (4.71 + 3.30i)T + (20.8 + 57.3i)T^{2} \)
67 \( 1 + (-1.89 + 5.19i)T + (-51.3 - 43.0i)T^{2} \)
71 \( 1 + (6.74 + 8.03i)T + (-12.3 + 69.9i)T^{2} \)
73 \( 1 - 4.72iT - 73T^{2} \)
79 \( 1 + (-5.96 - 12.7i)T + (-50.7 + 60.5i)T^{2} \)
83 \( 1 + (-11.2 - 1.97i)T + (77.9 + 28.3i)T^{2} \)
89 \( 1 + (5.09 + 2.37i)T + (57.2 + 68.1i)T^{2} \)
97 \( 1 + (-0.309 + 1.15i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.72209254345196358494252260814, −10.02365645956319350730931726252, −9.177956126990232658955507237817, −8.321395001504077670681811988523, −7.35671057682421965028574225570, −6.50932158918030155536158323569, −5.83867581768252846693972360491, −4.48004757456554392187967698463, −3.75329692724989981046601765795, −2.08351206386321628159675042440, 0.22964306331364436031938028571, 2.22707358254215702149217997133, 3.16488043863363606928252086946, 4.31549532993641149289436657835, 5.46167355843715384840458740520, 6.30714915368743356450811219461, 7.58481135663173512934574084062, 8.405296672283928636469841890573, 9.480164935579498637010334511778, 10.12892299800449118351099529290

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