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Properties

Label 2-666-111.89-c1-0-13
Degree $2$
Conductor $666$
Sign $-0.0787 + 0.996i$
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  + (0.906 − 0.422i)2-s + (0.642 − 0.766i)4-s + (1.52 − 1.06i)5-s + (0.800 − 4.53i)7-s + (0.258 − 0.965i)8-s + (0.928 − 1.60i)10-s + (−0.382 − 0.662i)11-s + (−5.11 + 0.447i)13-s + (−1.19 − 4.45i)14-s + (−0.173 − 0.984i)16-s + (−1.69 − 0.148i)17-s + (−2.29 + 4.91i)19-s + (0.161 − 1.85i)20-s + (−0.626 − 0.438i)22-s + (3.89 − 1.04i)23-s + ⋯
L(s)  = 1  + (0.640 − 0.298i)2-s + (0.321 − 0.383i)4-s + (0.680 − 0.476i)5-s + (0.302 − 1.71i)7-s + (0.0915 − 0.341i)8-s + (0.293 − 0.508i)10-s + (−0.115 − 0.199i)11-s + (−1.41 + 0.124i)13-s + (−0.318 − 1.18i)14-s + (−0.0434 − 0.246i)16-s + (−0.411 − 0.0360i)17-s + (−0.526 + 1.12i)19-s + (0.0362 − 0.413i)20-s + (−0.133 − 0.0935i)22-s + (0.812 − 0.217i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0787 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0787 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $-0.0787 + 0.996i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ -0.0787 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55195 - 1.67943i\)
\(L(\frac12)\) \(\approx\) \(1.55195 - 1.67943i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.906 + 0.422i)T \)
3 \( 1 \)
37 \( 1 + (1.35 + 5.92i)T \)
good5 \( 1 + (-1.52 + 1.06i)T + (1.71 - 4.69i)T^{2} \)
7 \( 1 + (-0.800 + 4.53i)T + (-6.57 - 2.39i)T^{2} \)
11 \( 1 + (0.382 + 0.662i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (5.11 - 0.447i)T + (12.8 - 2.25i)T^{2} \)
17 \( 1 + (1.69 + 0.148i)T + (16.7 + 2.95i)T^{2} \)
19 \( 1 + (2.29 - 4.91i)T + (-12.2 - 14.5i)T^{2} \)
23 \( 1 + (-3.89 + 1.04i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-9.12 - 2.44i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 + (-1.11 - 1.11i)T + 31iT^{2} \)
41 \( 1 + (-1.15 - 0.965i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (-5.40 + 5.40i)T - 43iT^{2} \)
47 \( 1 + (-4.91 - 2.83i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-8.79 + 1.55i)T + (49.8 - 18.1i)T^{2} \)
59 \( 1 + (-4.19 + 5.99i)T + (-20.1 - 55.4i)T^{2} \)
61 \( 1 + (-0.804 - 9.19i)T + (-60.0 + 10.5i)T^{2} \)
67 \( 1 + (-6.71 - 1.18i)T + (62.9 + 22.9i)T^{2} \)
71 \( 1 + (-2.79 - 7.69i)T + (-54.3 + 45.6i)T^{2} \)
73 \( 1 - 2.36iT - 73T^{2} \)
79 \( 1 + (3.40 + 4.86i)T + (-27.0 + 74.2i)T^{2} \)
83 \( 1 + (-6.85 - 8.17i)T + (-14.4 + 81.7i)T^{2} \)
89 \( 1 + (6.10 + 4.27i)T + (30.4 + 83.6i)T^{2} \)
97 \( 1 + (3.70 + 13.8i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.36332098618263210120514084197, −9.765472563353242683944228224037, −8.589989886390256067124629835635, −7.40991179161745468077452023986, −6.78637937933582202310264875340, −5.52227284539939894714850381486, −4.66944085953407985716526878316, −3.87805153787548813925459550551, −2.39984866893753045754499023491, −1.03419919929286992668558186106, 2.45713314452483452340558211455, 2.61109165212776709190954898776, 4.62840412558139533545458863248, 5.25359292759460066330914272954, 6.22962015309683349391835240844, 6.93631515025826892263838699342, 8.130348066347007630536164309572, 8.996941982612107203374050176519, 9.819373605039610764892003114735, 10.83556472507643457199978086882

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