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Properties

Label 2-666-111.23-c1-0-8
Degree $2$
Conductor $666$
Sign $-0.596 + 0.802i$
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 + (0.765 + 0.205i)5-s + (−0.896 − 1.55i)7-s + (−0.707 + 0.707i)8-s − 0.792·10-s − 4.98·11-s + (0.0278 − 0.103i)13-s + (1.26 + 1.26i)14-s + (0.500 − 0.866i)16-s + (−0.912 − 3.40i)17-s + (2.21 − 8.26i)19-s + (0.765 − 0.205i)20-s + (4.81 − 1.28i)22-s + (−5.99 + 5.99i)23-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.433 − 0.249i)4-s + (0.342 + 0.0917i)5-s + (−0.338 − 0.586i)7-s + (−0.249 + 0.249i)8-s − 0.250·10-s − 1.50·11-s + (0.00771 − 0.0288i)13-s + (0.338 + 0.338i)14-s + (0.125 − 0.216i)16-s + (−0.221 − 0.825i)17-s + (0.507 − 1.89i)19-s + (0.171 − 0.0458i)20-s + (1.02 − 0.275i)22-s + (−1.25 + 1.25i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.596 + 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.596 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $-0.596 + 0.802i$
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: \((2,\ 666,\ (\ :1/2),\ -0.596 + 0.802i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.210223 - 0.418349i\)
\(L(\frac12)\) \(\approx\) \(0.210223 - 0.418349i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.26 + 4.33i)T \)
good5 \( 1 + (-0.765 - 0.205i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (0.896 + 1.55i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 4.98T + 11T^{2} \)
13 \( 1 + (-0.0278 + 0.103i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (0.912 + 3.40i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-2.21 + 8.26i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (5.99 - 5.99i)T - 23iT^{2} \)
29 \( 1 + (-4.02 - 4.02i)T + 29iT^{2} \)
31 \( 1 + (6.76 - 6.76i)T - 31iT^{2} \)
41 \( 1 + (4.52 + 7.83i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.24 - 2.24i)T + 43iT^{2} \)
47 \( 1 + 8.63iT - 47T^{2} \)
53 \( 1 + (0.970 + 0.560i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.06 + 11.4i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (2.05 + 0.549i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (12.9 - 7.46i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.01 + 2.89i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 2.90iT - 73T^{2} \)
79 \( 1 + (0.0278 - 0.103i)T + (-68.4 - 39.5i)T^{2} \)
83 \( 1 + (3.41 + 1.97i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (5.75 - 1.54i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (0.841 + 0.841i)T + 97iT^{2} \)
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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

−10.14475766245502489506263664433, −9.438030909176713427386876283275, −8.529977867649191970344030541158, −7.40977173641161047004581187305, −7.04010843015363570092318208771, −5.71493307524564526864320323212, −4.91633457161076525210866664920, −3.30138377726715282018823405841, −2.16882569667906118512171019435, −0.28773467729686924643057342475, 1.85030474508787749550320152757, 2.86297965658433544969516051609, 4.24153435803341942386958947225, 5.80497386064732136358360082253, 6.11886538690077318257606243618, 7.86394575296528956683237349147, 7.988373462693170191291441121825, 9.207382031386169944913718387112, 10.08256183529808708824963935690, 10.44802954379204082631526895984

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