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Properties

Label 2-666-111.23-c1-0-3
Degree $2$
Conductor $666$
Sign $0.303 + 0.952i$
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  + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s + (−2.43 − 0.653i)5-s + (0.762 + 1.32i)7-s + (−0.707 + 0.707i)8-s + 2.52·10-s − 0.293·11-s + (0.472 − 1.76i)13-s + (−1.07 − 1.07i)14-s + (0.500 − 0.866i)16-s + (−0.0537 − 0.200i)17-s + (−0.213 + 0.798i)19-s + (−2.43 + 0.653i)20-s + (0.283 − 0.0760i)22-s + (5.10 − 5.10i)23-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.433 − 0.249i)4-s + (−1.09 − 0.292i)5-s + (0.288 + 0.498i)7-s + (−0.249 + 0.249i)8-s + 0.798·10-s − 0.0885·11-s + (0.130 − 0.488i)13-s + (−0.288 − 0.288i)14-s + (0.125 − 0.216i)16-s + (−0.0130 − 0.0486i)17-s + (−0.0490 + 0.183i)19-s + (−0.545 + 0.146i)20-s + (0.0604 − 0.0162i)22-s + (1.06 − 1.06i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.303 + 0.952i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.303 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $0.303 + 0.952i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (467, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ 0.303 + 0.952i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.559305 - 0.408727i\)
\(L(\frac12)\) \(\approx\) \(0.559305 - 0.408727i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
3 \( 1 \)
37 \( 1 + (1.03 + 5.99i)T \)
good5 \( 1 + (2.43 + 0.653i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (-0.762 - 1.32i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 0.293T + 11T^{2} \)
13 \( 1 + (-0.472 + 1.76i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (0.0537 + 0.200i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (0.213 - 0.798i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-5.10 + 5.10i)T - 23iT^{2} \)
29 \( 1 + (4.72 + 4.72i)T + 29iT^{2} \)
31 \( 1 + (-4.39 + 4.39i)T - 31iT^{2} \)
41 \( 1 + (-0.398 - 0.691i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.60 + 5.60i)T + 43iT^{2} \)
47 \( 1 + 0.508iT - 47T^{2} \)
53 \( 1 + (-3.09 - 1.78i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.715 + 2.66i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-0.820 - 0.219i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (0.106 - 0.0617i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-12.0 + 6.95i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 2.83iT - 73T^{2} \)
79 \( 1 + (0.472 - 1.76i)T + (-68.4 - 39.5i)T^{2} \)
83 \( 1 + (4.50 + 2.59i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-10.2 + 2.74i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (4.15 + 4.15i)T + 97iT^{2} \)
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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

−10.35872435341274170350673372111, −9.306713651021103979064132906154, −8.463312788125457688041010465672, −7.938549089875327835585198790612, −7.05531460054299966681853007737, −5.92166451684194461741917084335, −4.87485845996915324675911877695, −3.71532822936448876514746835835, −2.32418202777979566754875523348, −0.51065647968100563394450915000, 1.32872056518934918937361585757, 3.04633535496674423472631196249, 3.96430994709261158563589494416, 5.12305971132317264736093532639, 6.66425908565823427795030341512, 7.31322723007260712788954254090, 8.065616139312616558119625039683, 8.906869894919954405399172205881, 9.843362512054559224427098773862, 10.85220482066705555485726669561

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