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Properties

Label 2-222-37.33-c1-0-0
Degree $2$
Conductor $222$
Sign $-0.960 - 0.279i$
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  + (−0.173 + 0.984i)2-s + (−0.173 − 0.984i)3-s + (−0.939 − 0.342i)4-s + (−2.21 + 1.86i)5-s + 0.999·6-s + (−1.32 + 1.10i)7-s + (0.5 − 0.866i)8-s + (−0.939 + 0.342i)9-s + (−1.44 − 2.50i)10-s + (−1.81 + 3.14i)11-s + (−0.173 + 0.984i)12-s + (−1.33 − 0.486i)13-s + (−0.863 − 1.49i)14-s + (2.21 + 1.86i)15-s + (0.766 + 0.642i)16-s + (−5.06 + 1.84i)17-s + ⋯
L(s)  = 1  + (−0.122 + 0.696i)2-s + (−0.100 − 0.568i)3-s + (−0.469 − 0.171i)4-s + (−0.992 + 0.833i)5-s + 0.408·6-s + (−0.499 + 0.419i)7-s + (0.176 − 0.306i)8-s + (−0.313 + 0.114i)9-s + (−0.458 − 0.793i)10-s + (−0.547 + 0.949i)11-s + (−0.0501 + 0.284i)12-s + (−0.370 − 0.134i)13-s + (−0.230 − 0.399i)14-s + (0.573 + 0.480i)15-s + (0.191 + 0.160i)16-s + (−1.22 + 0.447i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 222 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 - 0.279i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 222 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.960 - 0.279i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(222\)    =    \(2 \cdot 3 \cdot 37\)
Sign: $-0.960 - 0.279i$
Analytic conductor: \(1.77267\)
Root analytic conductor: \(1.33141\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{222} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 222,\ (\ :1/2),\ -0.960 - 0.279i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0630229 + 0.441708i\)
\(L(\frac12)\) \(\approx\) \(0.0630229 + 0.441708i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.173 - 0.984i)T \)
3 \( 1 + (0.173 + 0.984i)T \)
37 \( 1 + (-3.41 + 5.03i)T \)
good5 \( 1 + (2.21 - 1.86i)T + (0.868 - 4.92i)T^{2} \)
7 \( 1 + (1.32 - 1.10i)T + (1.21 - 6.89i)T^{2} \)
11 \( 1 + (1.81 - 3.14i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.33 + 0.486i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (5.06 - 1.84i)T + (13.0 - 10.9i)T^{2} \)
19 \( 1 + (0.378 + 2.14i)T + (-17.8 + 6.49i)T^{2} \)
23 \( 1 + (-4.10 - 7.11i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.32 + 5.75i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.464T + 31T^{2} \)
41 \( 1 + (6.65 + 2.42i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 - 3.58T + 43T^{2} \)
47 \( 1 + (-1.24 - 2.15i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.68 - 3.09i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (-9.67 - 8.11i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-4.15 - 1.51i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (8.01 - 6.72i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-0.117 - 0.664i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + 2.27T + 73T^{2} \)
79 \( 1 + (3.45 - 2.90i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (15.7 - 5.72i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (11.6 + 9.81i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-7.74 - 13.4i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.80992287673897486834333478227, −11.77751238377052600187647384040, −10.86802183599372510123858884883, −9.679205627978880163378372922626, −8.527098817570206823392397119542, −7.35333324341842271276236984970, −7.01160591754625598976638282463, −5.72069618237432626952708307009, −4.26932896216873552331263944762, −2.65596632631464439864416899764, 0.38038686304675801180068713819, 3.00266743493904970698288069483, 4.21618274005767155109876899511, 5.05502864513987333505732599195, 6.78748555558673880588488145487, 8.314891439918182107196188357145, 8.808792524222499943842791460443, 10.05678295683016852892183498406, 10.91532387542298417208094553501, 11.70911856667981301122261886690

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