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Properties

Label 2-666-111.2-c1-0-2
Degree $2$
Conductor $666$
Sign $0.911 - 0.410i$
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 + (0.309 + 0.144i)5-s + (−2.12 + 0.774i)7-s + (−0.258 + 0.965i)8-s + (0.170 − 0.295i)10-s + (2.53 + 4.38i)11-s + (0.447 + 0.638i)13-s + (0.585 + 2.18i)14-s + (0.939 + 0.342i)16-s + (−3.48 + 4.97i)17-s + (5.68 − 0.497i)19-s + (−0.279 − 0.195i)20-s + (4.59 − 2.14i)22-s + (−2.40 + 0.644i)23-s + ⋯
L(s)  = 1  + (0.0616 − 0.704i)2-s + (−0.492 − 0.0868i)4-s + (0.138 + 0.0645i)5-s + (−0.804 + 0.292i)7-s + (−0.0915 + 0.341i)8-s + (0.0540 − 0.0935i)10-s + (0.763 + 1.32i)11-s + (0.123 + 0.177i)13-s + (0.156 + 0.584i)14-s + (0.234 + 0.0855i)16-s + (−0.844 + 1.20i)17-s + (1.30 − 0.114i)19-s + (−0.0625 − 0.0438i)20-s + (0.979 − 0.456i)22-s + (−0.501 + 0.134i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 - 0.410i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.911 - 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $0.911 - 0.410i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ 0.911 - 0.410i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23546 + 0.265483i\)
\(L(\frac12)\) \(\approx\) \(1.23546 + 0.265483i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.86 + 3.64i)T \)
good5 \( 1 + (-0.309 - 0.144i)T + (3.21 + 3.83i)T^{2} \)
7 \( 1 + (2.12 - 0.774i)T + (5.36 - 4.49i)T^{2} \)
11 \( 1 + (-2.53 - 4.38i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.447 - 0.638i)T + (-4.44 + 12.2i)T^{2} \)
17 \( 1 + (3.48 - 4.97i)T + (-5.81 - 15.9i)T^{2} \)
19 \( 1 + (-5.68 + 0.497i)T + (18.7 - 3.29i)T^{2} \)
23 \( 1 + (2.40 - 0.644i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-10.2 - 2.74i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 + (-4.79 - 4.79i)T + 31iT^{2} \)
41 \( 1 + (1.94 - 11.0i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (-5.24 + 5.24i)T - 43iT^{2} \)
47 \( 1 + (1.12 + 0.647i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.91 - 5.25i)T + (-40.6 - 34.0i)T^{2} \)
59 \( 1 + (0.911 + 1.95i)T + (-37.9 + 45.1i)T^{2} \)
61 \( 1 + (-2.57 + 1.80i)T + (20.8 - 57.3i)T^{2} \)
67 \( 1 + (1.75 + 4.82i)T + (-51.3 + 43.0i)T^{2} \)
71 \( 1 + (-6.66 + 7.94i)T + (-12.3 - 69.9i)T^{2} \)
73 \( 1 - 14.3iT - 73T^{2} \)
79 \( 1 + (1.92 - 4.12i)T + (-50.7 - 60.5i)T^{2} \)
83 \( 1 + (-4.84 + 0.854i)T + (77.9 - 28.3i)T^{2} \)
89 \( 1 + (8.92 - 4.16i)T + (57.2 - 68.1i)T^{2} \)
97 \( 1 + (2.93 + 10.9i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.37868897589705922521847660320, −9.858120429260397712693456263184, −9.101798266058131652434810523047, −8.174295250853543412225380260575, −6.84490760834768722482036212867, −6.20417958359619002507800332238, −4.84403820901641396260351259211, −3.94788571058268426120574610714, −2.78208800679078581382620237947, −1.54913406502899699944003134362, 0.71347511366293057615725441039, 2.94718090950977647422236647379, 3.91089761543468029652527967531, 5.14807527863920511030020654542, 6.14793287688057541097182183263, 6.74988266515317491813528299182, 7.78010705584479276824419446456, 8.736414284475320368075370233594, 9.469335000929845582066917613526, 10.21681921100327044676269369693

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