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Properties

Label 2-666-111.56-c1-0-9
Degree $2$
Conductor $666$
Sign $0.387 + 0.921i$
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 + (2.82 − 1.31i)5-s + (1.97 + 0.718i)7-s + (0.258 + 0.965i)8-s + (−1.55 − 2.70i)10-s + (1.33 − 2.31i)11-s + (−0.527 + 0.753i)13-s + (0.543 − 2.02i)14-s + (0.939 − 0.342i)16-s + (2.87 + 4.11i)17-s + (−2.92 − 0.256i)19-s + (−2.55 + 1.78i)20-s + (−2.42 − 1.13i)22-s + (2.70 + 0.725i)23-s + ⋯
L(s)  = 1  + (−0.0616 − 0.704i)2-s + (−0.492 + 0.0868i)4-s + (1.26 − 0.589i)5-s + (0.745 + 0.271i)7-s + (0.0915 + 0.341i)8-s + (−0.493 − 0.853i)10-s + (0.403 − 0.699i)11-s + (−0.146 + 0.208i)13-s + (0.145 − 0.541i)14-s + (0.234 − 0.0855i)16-s + (0.698 + 0.997i)17-s + (−0.671 − 0.0587i)19-s + (−0.571 + 0.399i)20-s + (−0.517 − 0.241i)22-s + (0.564 + 0.151i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.387 + 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.387 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $0.387 + 0.921i$
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: \((2,\ 666,\ (\ :1/2),\ 0.387 + 0.921i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56561 - 1.04030i\)
\(L(\frac12)\) \(\approx\) \(1.56561 - 1.04030i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.40 + 5.91i)T \)
good5 \( 1 + (-2.82 + 1.31i)T + (3.21 - 3.83i)T^{2} \)
7 \( 1 + (-1.97 - 0.718i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (-1.33 + 2.31i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.527 - 0.753i)T + (-4.44 - 12.2i)T^{2} \)
17 \( 1 + (-2.87 - 4.11i)T + (-5.81 + 15.9i)T^{2} \)
19 \( 1 + (2.92 + 0.256i)T + (18.7 + 3.29i)T^{2} \)
23 \( 1 + (-2.70 - 0.725i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-4.95 + 1.32i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (1.88 - 1.88i)T - 31iT^{2} \)
41 \( 1 + (-0.366 - 2.08i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (6.00 + 6.00i)T + 43iT^{2} \)
47 \( 1 + (5.03 - 2.90i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.29 + 6.30i)T + (-40.6 + 34.0i)T^{2} \)
59 \( 1 + (-5.91 + 12.6i)T + (-37.9 - 45.1i)T^{2} \)
61 \( 1 + (-5.22 - 3.66i)T + (20.8 + 57.3i)T^{2} \)
67 \( 1 + (2.91 - 8.01i)T + (-51.3 - 43.0i)T^{2} \)
71 \( 1 + (3.95 + 4.70i)T + (-12.3 + 69.9i)T^{2} \)
73 \( 1 - 9.01iT - 73T^{2} \)
79 \( 1 + (-0.222 - 0.477i)T + (-50.7 + 60.5i)T^{2} \)
83 \( 1 + (-3.14 - 0.553i)T + (77.9 + 28.3i)T^{2} \)
89 \( 1 + (2.30 + 1.07i)T + (57.2 + 68.1i)T^{2} \)
97 \( 1 + (3.13 - 11.7i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.33265028025473195684854587540, −9.575171353809628216111936808074, −8.704686625731457118749532313132, −8.202780556715153584199118587787, −6.61141629423193906764529125501, −5.62193150200311203186662534628, −4.92318160071648507152453236569, −3.64827434496443977905683515959, −2.19456359278047186289657695644, −1.30070275762464398403898104830, 1.53153405760086804788538302080, 2.89932060180985196214627727927, 4.53305569734583568005297171668, 5.30146397779020743162623120981, 6.35091290453543996089214000968, 7.02414480565474753880706727354, 7.937292644071499692821751439151, 8.982929162523023011411829650493, 9.853452071277632350371678865646, 10.35937220729102904450986140745

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