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Properties

Label 2-220-11.2-c2-0-3
Degree $2$
Conductor $220$
Sign $0.976 - 0.217i$
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  + (−3.86 + 2.80i)3-s + (−0.690 − 2.12i)5-s + (5.14 − 7.08i)7-s + (4.27 − 13.1i)9-s + (−4.15 + 10.1i)11-s + (−6.21 − 2.02i)13-s + (8.64 + 6.28i)15-s + (30.3 − 9.85i)17-s + (15.4 + 21.3i)19-s + 41.8i·21-s + 41.5·23-s + (−4.04 + 2.93i)25-s + (7.16 + 22.0i)27-s + (10.2 − 14.1i)29-s + (−4.59 + 14.1i)31-s + ⋯
L(s)  = 1  + (−1.28 + 0.936i)3-s + (−0.138 − 0.425i)5-s + (0.735 − 1.01i)7-s + (0.475 − 1.46i)9-s + (−0.377 + 0.926i)11-s + (−0.478 − 0.155i)13-s + (0.576 + 0.418i)15-s + (1.78 − 0.579i)17-s + (0.814 + 1.12i)19-s + 1.99i·21-s + 1.80·23-s + (−0.161 + 0.117i)25-s + (0.265 + 0.816i)27-s + (0.355 − 0.488i)29-s + (−0.148 + 0.456i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.217i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.976 - 0.217i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $0.976 - 0.217i$
Analytic conductor: \(5.99456\)
Root analytic conductor: \(2.44838\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{220} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 220,\ (\ :1),\ 0.976 - 0.217i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.04639 + 0.115174i\)
\(L(\frac12)\) \(\approx\) \(1.04639 + 0.115174i\)
\(L(2)\) 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 \( 1 + (0.690 + 2.12i)T \)
11 \( 1 + (4.15 - 10.1i)T \)
good3 \( 1 + (3.86 - 2.80i)T + (2.78 - 8.55i)T^{2} \)
7 \( 1 + (-5.14 + 7.08i)T + (-15.1 - 46.6i)T^{2} \)
13 \( 1 + (6.21 + 2.02i)T + (136. + 99.3i)T^{2} \)
17 \( 1 + (-30.3 + 9.85i)T + (233. - 169. i)T^{2} \)
19 \( 1 + (-15.4 - 21.3i)T + (-111. + 343. i)T^{2} \)
23 \( 1 - 41.5T + 529T^{2} \)
29 \( 1 + (-10.2 + 14.1i)T + (-259. - 799. i)T^{2} \)
31 \( 1 + (4.59 - 14.1i)T + (-777. - 564. i)T^{2} \)
37 \( 1 + (27.3 + 19.8i)T + (423. + 1.30e3i)T^{2} \)
41 \( 1 + (-2.22 - 3.05i)T + (-519. + 1.59e3i)T^{2} \)
43 \( 1 - 23.8iT - 1.84e3T^{2} \)
47 \( 1 + (-20.2 + 14.7i)T + (682. - 2.10e3i)T^{2} \)
53 \( 1 + (-32.5 + 100. i)T + (-2.27e3 - 1.65e3i)T^{2} \)
59 \( 1 + (49.2 + 35.8i)T + (1.07e3 + 3.31e3i)T^{2} \)
61 \( 1 + (-76.1 + 24.7i)T + (3.01e3 - 2.18e3i)T^{2} \)
67 \( 1 - 109.T + 4.48e3T^{2} \)
71 \( 1 + (10.4 + 32.2i)T + (-4.07e3 + 2.96e3i)T^{2} \)
73 \( 1 + (23.7 - 32.6i)T + (-1.64e3 - 5.06e3i)T^{2} \)
79 \( 1 + (-66.5 - 21.6i)T + (5.04e3 + 3.66e3i)T^{2} \)
83 \( 1 + (42.8 - 13.9i)T + (5.57e3 - 4.04e3i)T^{2} \)
89 \( 1 + 24.9T + 7.92e3T^{2} \)
97 \( 1 + (39.7 - 122. i)T + (-7.61e3 - 5.53e3i)T^{2} \)
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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

−11.95366024008989124390342084257, −11.08249377593587570047963483225, −10.16016762046936389002123462122, −9.676689244531214083443559310672, −7.927389108728491040233635248346, −7.05163233109855889502658889863, −5.28583132092568521448431126958, −5.00313236478724114101438289257, −3.71597062003659372104514192025, −0.956457154888133134243995279363, 1.07605229318577362977115656604, 2.87116620930587495813513111281, 5.19336268760880421288537122566, 5.64209396336760779783072662277, 6.91041208190727031768849528814, 7.76184429581966907724568616035, 8.945833162779363040016513084027, 10.50568298852932370527583487736, 11.31059047589562566947225174744, 11.92546601200809301971757951529

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