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Properties

Label 2-666-111.14-c1-0-3
Degree $2$
Conductor $666$
Sign $0.690 + 0.723i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.690 + 0.723i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.690 + 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $0.690 + 0.723i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ 0.690 + 0.723i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16989 - 0.500493i\)
\(L(\frac12)\) \(\approx\) \(1.16989 - 0.500493i\)
\(L(\frac{3}{2})\) 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

−10.28020140430667721634388690958, −9.807198582230523127516659484554, −8.941541239992399894596607486875, −7.55147404740296816803573963378, −7.18852166355302504907081719082, −5.93760924257146085989573322573, −4.62220237102513721150247863720, −3.55054841379858594156274084274, −2.75362200415880039695538384352, −0.989498649154790071725286313333, 1.12411464036829486731155354282, 2.98911111113836348909812297716, 4.42877417764614220942241956939, 5.17515251131172490298000152759, 6.26414451101063990667984611693, 7.03663564934327852840248745824, 8.119107045194168839773980726905, 8.889177597864080382002626113214, 9.495840932497176271964077507121, 10.34233733678853363130670933125

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