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Properties

Label 2-222-37.11-c1-0-5
Degree $2$
Conductor $222$
Sign $0.0405 + 0.999i$
Analytic cond. $1.77267$
Root an. cond. $1.33141$
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  + (0.866 − 0.5i)2-s + (0.5 − 0.866i)3-s + (0.499 − 0.866i)4-s + (−2.05 − 1.18i)5-s − 0.999i·6-s + (0.762 − 1.32i)7-s − 0.999i·8-s + (−0.499 − 0.866i)9-s − 2.37·10-s − 0.152·11-s + (−0.499 − 0.866i)12-s + (2.41 + 1.39i)13-s − 1.52i·14-s + (−2.05 + 1.18i)15-s + (−0.5 − 0.866i)16-s + (0.454 − 0.262i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.288 − 0.499i)3-s + (0.249 − 0.433i)4-s + (−0.918 − 0.530i)5-s − 0.408i·6-s + (0.288 − 0.498i)7-s − 0.353i·8-s + (−0.166 − 0.288i)9-s − 0.750·10-s − 0.0458·11-s + (−0.144 − 0.249i)12-s + (0.670 + 0.387i)13-s − 0.407i·14-s + (−0.530 + 0.306i)15-s + (−0.125 − 0.216i)16-s + (0.110 − 0.0635i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 222 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0405 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 222 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0405 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(222\)    =    \(2 \cdot 3 \cdot 37\)
Sign: $0.0405 + 0.999i$
Analytic conductor: \(1.77267\)
Root analytic conductor: \(1.33141\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{222} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 222,\ (\ :1/2),\ 0.0405 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18601 - 1.13891i\)
\(L(\frac12)\) \(\approx\) \(1.18601 - 1.13891i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-1.62 - 5.86i)T \)
good5 \( 1 + (2.05 + 1.18i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.762 + 1.32i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 0.152T + 11T^{2} \)
13 \( 1 + (-2.41 - 1.39i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.454 + 0.262i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.24 - 2.44i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 4.63iT - 23T^{2} \)
29 \( 1 - 2.90iT - 29T^{2} \)
31 \( 1 - 1.35iT - 31T^{2} \)
41 \( 1 + (-2.91 + 5.05i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 6.73iT - 43T^{2} \)
47 \( 1 + 5.72T + 47T^{2} \)
53 \( 1 + (1.57 + 2.73i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.21 - 4.16i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.35 - 4.24i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.92 - 10.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.29 - 3.97i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 7.25T + 73T^{2} \)
79 \( 1 + (11.2 + 6.52i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.31 + 10.9i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-11.2 + 6.48i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 13.9iT - 97T^{2} \)
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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

−11.95247528464082996447430646712, −11.50267512963877959832400846100, −10.28995298121544767319628102049, −8.986275523529901075949016823414, −7.928372675580627076123534602363, −7.08435515615153789235133579040, −5.64943648594141155416074974085, −4.33312276197344953985978808264, −3.34310575318121975569092144670, −1.34663943289916381856143891189, 2.82364129114850005660600739922, 3.88479939001098214446704215533, 5.05270271261214440754000483726, 6.29292055548117074759491575912, 7.59452097608398055317865900771, 8.305125750935129326450353924347, 9.511369845996772087193604706232, 10.91440255899779608170346919159, 11.46552578967196701688389302201, 12.51322851171719546844412489065

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