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Properties

Label 2-220-5.3-c2-0-5
Degree $2$
Conductor $220$
Sign $0.896 - 0.443i$
Analytic cond. $5.99456$
Root an. cond. $2.44838$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.36 + 1.36i)3-s + (4.24 + 2.64i)5-s + (4.16 − 4.16i)7-s − 5.29i·9-s + 3.31·11-s + (7.53 + 7.53i)13-s + (2.16 + 9.37i)15-s + (−10.2 + 10.2i)17-s + 6.46i·19-s + 11.3·21-s + (−0.810 − 0.810i)23-s + (10.9 + 22.4i)25-s + (19.4 − 19.4i)27-s − 2.81i·29-s + 13.1·31-s + ⋯
L(s)  = 1  + (0.453 + 0.453i)3-s + (0.848 + 0.529i)5-s + (0.594 − 0.594i)7-s − 0.588i·9-s + 0.301·11-s + (0.579 + 0.579i)13-s + (0.144 + 0.625i)15-s + (−0.605 + 0.605i)17-s + 0.340i·19-s + 0.539·21-s + (−0.0352 − 0.0352i)23-s + (0.439 + 0.898i)25-s + (0.720 − 0.720i)27-s − 0.0969i·29-s + 0.424·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.896 - 0.443i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.896 - 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $0.896 - 0.443i$
Analytic conductor: \(5.99456\)
Root analytic conductor: \(2.44838\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{220} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 220,\ (\ :1),\ 0.896 - 0.443i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.11360 + 0.493956i\)
\(L(\frac12)\) \(\approx\) \(2.11360 + 0.493956i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-4.24 - 2.64i)T \)
11 \( 1 - 3.31T \)
good3 \( 1 + (-1.36 - 1.36i)T + 9iT^{2} \)
7 \( 1 + (-4.16 + 4.16i)T - 49iT^{2} \)
13 \( 1 + (-7.53 - 7.53i)T + 169iT^{2} \)
17 \( 1 + (10.2 - 10.2i)T - 289iT^{2} \)
19 \( 1 - 6.46iT - 361T^{2} \)
23 \( 1 + (0.810 + 0.810i)T + 529iT^{2} \)
29 \( 1 + 2.81iT - 841T^{2} \)
31 \( 1 - 13.1T + 961T^{2} \)
37 \( 1 + (-4.46 + 4.46i)T - 1.36e3iT^{2} \)
41 \( 1 + 71.2T + 1.68e3T^{2} \)
43 \( 1 + (17.9 + 17.9i)T + 1.84e3iT^{2} \)
47 \( 1 + (-9.43 + 9.43i)T - 2.20e3iT^{2} \)
53 \( 1 + (-1.65 - 1.65i)T + 2.80e3iT^{2} \)
59 \( 1 + 35.5iT - 3.48e3T^{2} \)
61 \( 1 + 71.8T + 3.72e3T^{2} \)
67 \( 1 + (-64.1 + 64.1i)T - 4.48e3iT^{2} \)
71 \( 1 + 77.7T + 5.04e3T^{2} \)
73 \( 1 + (66.9 + 66.9i)T + 5.32e3iT^{2} \)
79 \( 1 + 42.5iT - 6.24e3T^{2} \)
83 \( 1 + (58.8 + 58.8i)T + 6.88e3iT^{2} \)
89 \( 1 - 111. iT - 7.92e3T^{2} \)
97 \( 1 + (-52.0 + 52.0i)T - 9.40e3iT^{2} \)
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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

−12.03096713513333828552122373817, −10.96172295408158773506846073458, −10.18000420095989614889444589553, −9.246885380984771035009134053465, −8.362636117114452423973286268437, −6.92114774608541240208634202296, −6.07309038589560335904590209274, −4.49083520960824496796110861056, −3.38234638682046916073462839153, −1.69541760760645339835147635779, 1.52140200820051163532608095565, 2.70981712925565931253065485483, 4.72692274548705372309127150574, 5.66371584154087404615200265276, 6.93699809445567701137687126172, 8.285104464577923062972439351834, 8.779683725577935127124302569220, 9.950209789123311984620734654781, 11.06590080723307797294134684875, 12.08661754246806697987948425179

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