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Properties

Label 2-222-37.27-c1-0-5
Degree $2$
Conductor $222$
Sign $0.835 - 0.549i$
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 + (2.92 − 1.68i)5-s + 0.999i·6-s + (−0.896 − 1.55i)7-s + 0.999i·8-s + (−0.499 + 0.866i)9-s + 3.37·10-s − 2.57·11-s + (−0.499 + 0.866i)12-s + (−0.454 + 0.262i)13-s − 1.79i·14-s + (2.92 + 1.68i)15-s + (−0.5 + 0.866i)16-s + (−2.41 − 1.39i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.288 + 0.499i)3-s + (0.249 + 0.433i)4-s + (1.30 − 0.754i)5-s + 0.408i·6-s + (−0.338 − 0.586i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + 1.06·10-s − 0.777·11-s + (−0.144 + 0.249i)12-s + (−0.125 + 0.0727i)13-s − 0.479i·14-s + (0.754 + 0.435i)15-s + (−0.125 + 0.216i)16-s + (−0.586 − 0.338i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 222 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 - 0.549i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 222 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.835 - 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(222\)    =    \(2 \cdot 3 \cdot 37\)
Sign: $0.835 - 0.549i$
Analytic conductor: \(1.77267\)
Root analytic conductor: \(1.33141\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{222} (175, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 222,\ (\ :1/2),\ 0.835 - 0.549i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.90277 + 0.570051i\)
\(L(\frac12)\) \(\approx\) \(1.90277 + 0.570051i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.70 + 2.10i)T \)
good5 \( 1 + (-2.92 + 1.68i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.896 + 1.55i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 2.57T + 11T^{2} \)
13 \( 1 + (0.454 - 0.262i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.41 + 1.39i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.60 - 2.08i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 8.63iT - 23T^{2} \)
29 \( 1 - 2.83iT - 29T^{2} \)
31 \( 1 + 7.10iT - 31T^{2} \)
41 \( 1 + (-0.0459 - 0.0795i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 12.2iT - 43T^{2} \)
47 \( 1 + 9.93T + 47T^{2} \)
53 \( 1 + (-0.847 + 1.46i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-8.48 - 4.89i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.36 - 1.94i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.79 - 8.30i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.66 - 8.07i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 8.14T + 73T^{2} \)
79 \( 1 + (-0.964 + 0.556i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.316 + 0.548i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (13.0 + 7.54i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 2.45iT - 97T^{2} \)
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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

−12.88837320713595734312364959442, −11.41339626952706864030045220892, −10.21991180499293575135688652983, −9.525345068790395151359591990139, −8.470120457463293036445122137713, −7.21744720341223400895839085533, −5.88188113211368006775573692587, −5.11010917872700980100245423522, −3.87167519756698446249581856734, −2.22046353861247514900630381766, 2.19762243688025543700886672943, 2.86773865785060594791987754438, 4.84402400493868485617287047830, 6.22115301669426470985675049549, 6.58834835383159179231013756660, 8.256407773258243723594498774467, 9.447037591972628260668052724097, 10.37164810695218153590505903868, 11.15070589408625446354642946944, 12.64209893487852632734649131332

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