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Properties

Label 2-220-220.219-c1-0-7
Degree $2$
Conductor $220$
Sign $0.829 - 0.559i$
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  + (−1.24 − 0.664i)2-s + (1.11 + 1.65i)4-s + 2.23·5-s + 4.29i·7-s + (−0.294 − 2.81i)8-s − 3·9-s + (−2.79 − 1.48i)10-s + 3.31i·11-s − 1.90·13-s + (2.85 − 5.36i)14-s + (−1.49 + 3.70i)16-s + 8.08·17-s + (3.74 + 1.99i)18-s + (2.50 + 3.70i)20-s + (2.20 − 4.14i)22-s + ⋯
L(s)  = 1  + (−0.882 − 0.469i)2-s + (0.559 + 0.829i)4-s + 0.999·5-s + 1.62i·7-s + (−0.104 − 0.994i)8-s − 9-s + (−0.882 − 0.469i)10-s + 1.00i·11-s − 0.529·13-s + (0.762 − 1.43i)14-s + (−0.374 + 0.927i)16-s + 1.95·17-s + (0.882 + 0.469i)18-s + (0.559 + 0.829i)20-s + (0.469 − 0.882i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.829 - 0.559i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.829 - 0.559i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $0.829 - 0.559i$
Analytic conductor: \(1.75670\)
Root analytic conductor: \(1.32540\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{220} (219, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 220,\ (\ :1/2),\ 0.829 - 0.559i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.851941 + 0.260365i\)
\(L(\frac12)\) \(\approx\) \(0.851941 + 0.260365i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.24 + 0.664i)T \)
5 \( 1 - 2.23T \)
11 \( 1 - 3.31iT \)
good3 \( 1 + 3T^{2} \)
7 \( 1 - 4.29iT - 7T^{2} \)
13 \( 1 + 1.90T + 13T^{2} \)
17 \( 1 - 8.08T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 6.63iT - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 1.01iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 14.8iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 - 14.8iT - 71T^{2} \)
73 \( 1 + 11.8T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 18.2iT - 83T^{2} \)
89 \( 1 - 13.4T + 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−12.19430123383967887211802677783, −11.51799079167151327950356157431, −10.10044111731562785260228317422, −9.538631394086754896917215038881, −8.701017760776357036292952882274, −7.65089402091915618464691992698, −6.16756336670780461942788990235, −5.29526283324559675778006604176, −2.96207303717947277189319206807, −2.02881335765112939592100144208, 1.06519132048910787268098613289, 3.10735027299641714476863282106, 5.23479218386229192692071262718, 6.12772319146569543933992595316, 7.26118771578290698805839930655, 8.193588181776631068815992469664, 9.294705740015931014339870369382, 10.28908412523735357603202407418, 10.73267943137480704734415421868, 11.96858282229378580929994036018

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