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Properties

Label 2-666-111.8-c1-0-1
Degree $2$
Conductor $666$
Sign $-0.451 - 0.892i$
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  + (0.258 − 0.965i)2-s + (−0.866 − 0.499i)4-s + (0.448 + 1.67i)5-s + (−1 + 1.73i)7-s + (−0.707 + 0.707i)8-s + 1.73·10-s − 3.86·11-s + (−4.09 + 1.09i)13-s + (1.41 + 1.41i)14-s + (0.500 + 0.866i)16-s + (−5.53 − 1.48i)17-s + (1 − 0.267i)19-s + (0.448 − 1.67i)20-s + (−0.999 + 3.73i)22-s + (−3.86 + 3.86i)23-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (−0.433 − 0.249i)4-s + (0.200 + 0.748i)5-s + (−0.377 + 0.654i)7-s + (−0.249 + 0.249i)8-s + 0.547·10-s − 1.16·11-s + (−1.13 + 0.304i)13-s + (0.377 + 0.377i)14-s + (0.125 + 0.216i)16-s + (−1.34 − 0.359i)17-s + (0.229 − 0.0614i)19-s + (0.100 − 0.374i)20-s + (−0.213 + 0.795i)22-s + (−0.805 + 0.805i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.451 - 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.451 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $-0.451 - 0.892i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ -0.451 - 0.892i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.248409 + 0.404196i\)
\(L(\frac12)\) \(\approx\) \(0.248409 + 0.404196i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 + 0.965i)T \)
3 \( 1 \)
37 \( 1 + (-2.59 - 5.5i)T \)
good5 \( 1 + (-0.448 - 1.67i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (1 - 1.73i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 3.86T + 11T^{2} \)
13 \( 1 + (4.09 - 1.09i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (5.53 + 1.48i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1 + 0.267i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (3.86 - 3.86i)T - 23iT^{2} \)
29 \( 1 + (6.50 + 6.50i)T + 29iT^{2} \)
31 \( 1 + (-1.26 + 1.26i)T - 31iT^{2} \)
41 \( 1 + (0.258 - 0.448i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.73 - 4.73i)T + 43iT^{2} \)
47 \( 1 - 5.93iT - 47T^{2} \)
53 \( 1 + (1.22 - 0.707i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-9.14 - 2.44i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (2.42 + 9.06i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-9 - 5.19i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-7.58 - 4.38i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 4iT - 73T^{2} \)
79 \( 1 + (13.9 - 3.73i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (-4.89 + 2.82i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.258 - 0.965i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-3.63 - 3.63i)T + 97iT^{2} \)
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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

−10.94232121227618121697185837855, −9.777609936611542216230003239508, −9.566131017100761297077365885239, −8.242416071836286633490344844390, −7.27977117844387116269253459233, −6.22303450705351944778312362552, −5.29103464384425615629978052339, −4.23681771766712760166906858290, −2.74453150276345916783125541378, −2.33014379276947866152203052626, 0.21707816615006238682806785175, 2.35959872363168242656920496229, 3.87654230375510779276845572926, 4.90228498266116389687364350223, 5.55689832140881628385575390259, 6.80993900190998170203849368283, 7.49420402443323645644738760986, 8.440781442011797626300551912092, 9.232462951138708115281037209270, 10.18672244702866431084248007265

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