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Properties

Label 2-220-44.19-c1-0-17
Degree $2$
Conductor $220$
Sign $0.122 + 0.992i$
Analytic cond. $1.75670$
Root an. cond. $1.32540$
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  + (−0.317 − 1.37i)2-s + (2.51 − 0.817i)3-s + (−1.79 + 0.874i)4-s + (0.809 + 0.587i)5-s + (−1.92 − 3.21i)6-s + (−0.0373 + 0.114i)7-s + (1.77 + 2.20i)8-s + (3.24 − 2.35i)9-s + (0.553 − 1.30i)10-s + (2.16 − 2.50i)11-s + (−3.81 + 3.67i)12-s + (−3.60 − 4.95i)13-s + (0.170 + 0.0150i)14-s + (2.51 + 0.817i)15-s + (2.47 − 3.14i)16-s + (−3.69 + 5.08i)17-s + ⋯
L(s)  = 1  + (−0.224 − 0.974i)2-s + (1.45 − 0.472i)3-s + (−0.899 + 0.437i)4-s + (0.361 + 0.262i)5-s + (−0.786 − 1.31i)6-s + (−0.0141 + 0.0434i)7-s + (0.627 + 0.778i)8-s + (1.08 − 0.784i)9-s + (0.175 − 0.411i)10-s + (0.654 − 0.756i)11-s + (−1.10 + 1.05i)12-s + (−0.998 − 1.37i)13-s + (0.0454 + 0.00401i)14-s + (0.649 + 0.211i)15-s + (0.617 − 0.786i)16-s + (−0.895 + 1.23i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.122 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.122 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $0.122 + 0.992i$
Analytic conductor: \(1.75670\)
Root analytic conductor: \(1.32540\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{220} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 220,\ (\ :1/2),\ 0.122 + 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19675 - 1.05796i\)
\(L(\frac12)\) \(\approx\) \(1.19675 - 1.05796i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.317 + 1.37i)T \)
5 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (-2.16 + 2.50i)T \)
good3 \( 1 + (-2.51 + 0.817i)T + (2.42 - 1.76i)T^{2} \)
7 \( 1 + (0.0373 - 0.114i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (3.60 + 4.95i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (3.69 - 5.08i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (-2.12 - 6.54i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 0.892iT - 23T^{2} \)
29 \( 1 + (8.01 + 2.60i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-2.18 - 3.00i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.0956 + 0.294i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.65 + 0.538i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 - 2.73T + 43T^{2} \)
47 \( 1 + (0.176 - 0.0572i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (5.28 - 3.83i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.79 - 0.583i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (-0.785 + 1.08i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + 10.2iT - 67T^{2} \)
71 \( 1 + (1.37 - 1.89i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.938 - 0.305i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-3.76 + 2.73i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (4.72 + 3.43i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 12.6T + 89T^{2} \)
97 \( 1 + (-4.90 + 3.56i)T + (29.9 - 92.2i)T^{2} \)
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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

−12.32599199729321091847118568278, −10.96123129576630163621910689921, −10.00992204999340640410827275038, −9.159054515339040282234077762175, −8.246442008997446956553635520674, −7.55515754611158899479011044035, −5.78642745733490598131442511975, −3.86920646866479806392111468243, −2.94736250963279701388658599669, −1.74163957948841734115940594688, 2.29673661208280075467558209571, 4.17464197974398390641968060244, 4.93979583381904405025878302397, 6.84231427428518995568195784674, 7.41350583842732264545700507194, 8.890159734195619177003032644500, 9.283975136801755883764257174992, 9.803658319139354316446761733422, 11.51439553018973332085834794095, 13.02249218893413163287858984033

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