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Properties

Label 2-220-5.4-c5-0-21
Degree $2$
Conductor $220$
Sign $0.228 + 0.973i$
Analytic cond. $35.2844$
Root an. cond. $5.94007$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 26.2i·3-s + (12.7 + 54.4i)5-s − 64.0i·7-s − 443.·9-s + 121·11-s − 528. i·13-s + (−1.42e3 + 335. i)15-s − 1.54e3i·17-s − 3.10e3·19-s + 1.67e3·21-s − 1.78e3i·23-s + (−2.79e3 + 1.39e3i)25-s − 5.26e3i·27-s − 4.89e3·29-s + 1.17e3·31-s + ⋯
L(s)  = 1  + 1.68i·3-s + (0.228 + 0.973i)5-s − 0.494i·7-s − 1.82·9-s + 0.301·11-s − 0.867i·13-s + (−1.63 + 0.384i)15-s − 1.29i·17-s − 1.97·19-s + 0.831·21-s − 0.705i·23-s + (−0.895 + 0.445i)25-s − 1.38i·27-s − 1.08·29-s + 0.219·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.228 + 0.973i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.228 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $0.228 + 0.973i$
Analytic conductor: \(35.2844\)
Root analytic conductor: \(5.94007\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{220} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 220,\ (\ :5/2),\ 0.228 + 0.973i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.1257991912\)
\(L(\frac12)\) \(\approx\) \(0.1257991912\)
\(L(\frac{7}{2})\) 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 \( 1 + (-12.7 - 54.4i)T \)
11 \( 1 - 121T \)
good3 \( 1 - 26.2iT - 243T^{2} \)
7 \( 1 + 64.0iT - 1.68e4T^{2} \)
13 \( 1 + 528. iT - 3.71e5T^{2} \)
17 \( 1 + 1.54e3iT - 1.41e6T^{2} \)
19 \( 1 + 3.10e3T + 2.47e6T^{2} \)
23 \( 1 + 1.78e3iT - 6.43e6T^{2} \)
29 \( 1 + 4.89e3T + 2.05e7T^{2} \)
31 \( 1 - 1.17e3T + 2.86e7T^{2} \)
37 \( 1 - 1.26e4iT - 6.93e7T^{2} \)
41 \( 1 - 1.52e3T + 1.15e8T^{2} \)
43 \( 1 + 1.15e4iT - 1.47e8T^{2} \)
47 \( 1 + 1.97e4iT - 2.29e8T^{2} \)
53 \( 1 - 1.62e4iT - 4.18e8T^{2} \)
59 \( 1 + 3.94e4T + 7.14e8T^{2} \)
61 \( 1 - 1.88e4T + 8.44e8T^{2} \)
67 \( 1 - 4.04e4iT - 1.35e9T^{2} \)
71 \( 1 + 6.52e4T + 1.80e9T^{2} \)
73 \( 1 + 4.63e4iT - 2.07e9T^{2} \)
79 \( 1 - 2.01e4T + 3.07e9T^{2} \)
83 \( 1 - 8.59e4iT - 3.93e9T^{2} \)
89 \( 1 + 1.20e4T + 5.58e9T^{2} \)
97 \( 1 + 1.37e5iT - 8.58e9T^{2} \)
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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

−10.71884116915888744290315104296, −10.44439392925725388534174089090, −9.518127325479863721122990604124, −8.483923255494046631432929848002, −7.06072265045491266602158159483, −5.87291305896101077764680554704, −4.65517925568218560936373972014, −3.68356943561501691654136554155, −2.60713839568317739207525011394, −0.03508440682371567314721820898, 1.49419607824154743958717521482, 2.12613273853625183774241182471, 4.18433960833391905576088502868, 5.81034520946496049619229421086, 6.41420536354077558026002965468, 7.67513888258630631453653702941, 8.556786906066278736230702814719, 9.258182914359376951987978095431, 10.95888388135685110107767865941, 11.97139858534285079010151998906

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