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Properties

Label 2-220-44.19-c1-0-10
Degree $2$
Conductor $220$
Sign $0.785 + 0.619i$
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  + (−1.21 − 0.727i)2-s + (0.919 − 0.298i)3-s + (0.941 + 1.76i)4-s + (0.809 + 0.587i)5-s + (−1.33 − 0.306i)6-s + (0.819 − 2.52i)7-s + (0.141 − 2.82i)8-s + (−1.67 + 1.21i)9-s + (−0.553 − 1.30i)10-s + (3.31 − 0.0962i)11-s + (1.39 + 1.34i)12-s + (3.18 + 4.38i)13-s + (−2.82 + 2.46i)14-s + (0.919 + 0.298i)15-s + (−2.22 + 3.32i)16-s + (2.85 − 3.92i)17-s + ⋯
L(s)  = 1  + (−0.857 − 0.514i)2-s + (0.530 − 0.172i)3-s + (0.470 + 0.882i)4-s + (0.361 + 0.262i)5-s + (−0.543 − 0.125i)6-s + (0.309 − 0.953i)7-s + (0.0501 − 0.998i)8-s + (−0.557 + 0.404i)9-s + (−0.175 − 0.411i)10-s + (0.999 − 0.0290i)11-s + (0.401 + 0.387i)12-s + (0.884 + 1.21i)13-s + (−0.756 + 0.658i)14-s + (0.237 + 0.0771i)15-s + (−0.556 + 0.830i)16-s + (0.692 − 0.952i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.785 + 0.619i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.785 + 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $0.785 + 0.619i$
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: \((2,\ 220,\ (\ :1/2),\ 0.785 + 0.619i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02350 - 0.355125i\)
\(L(\frac12)\) \(\approx\) \(1.02350 - 0.355125i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.21 + 0.727i)T \)
5 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (-3.31 + 0.0962i)T \)
good3 \( 1 + (-0.919 + 0.298i)T + (2.42 - 1.76i)T^{2} \)
7 \( 1 + (-0.819 + 2.52i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-3.18 - 4.38i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-2.85 + 3.92i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.860 + 2.64i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 6.73iT - 23T^{2} \)
29 \( 1 + (5.27 + 1.71i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-2.76 - 3.81i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.07 - 6.38i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (5.13 - 1.66i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 - 5.61T + 43T^{2} \)
47 \( 1 + (11.2 - 3.66i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-4.06 + 2.95i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (3.91 + 1.27i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (4.69 - 6.46i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + 1.99iT - 67T^{2} \)
71 \( 1 + (7.82 - 10.7i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (10.0 + 3.26i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (9.25 - 6.72i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (7.51 + 5.45i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 14.5T + 89T^{2} \)
97 \( 1 + (-2.55 + 1.85i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.77630045264357452672409503736, −11.20451168176233635852199506280, −10.22161078704056332850317185354, −9.156550160402639284822714148223, −8.469596878807816901635015270290, −7.29199475759097600807741301574, −6.47584689261077473610732859057, −4.34396262172962019733651815047, −3.01546319558023640236224805950, −1.50842593829999927508949326669, 1.67768487296800924983497055325, 3.46752358364003216049884494716, 5.64760165041214599391904731928, 6.01258246756491939269692195448, 7.74686810386392356856377071795, 8.592443811454467566825596151921, 9.168087121596435532768775112872, 10.13325328496421396659319767848, 11.30492544636048388654637106616, 12.20955779200205449296141618888

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