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Properties

Label 2-220-44.43-c1-0-8
Degree $2$
Conductor $220$
Sign $0.904 - 0.426i$
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  + 1.41i·2-s − 1.41i·3-s − 2.00·4-s − 5-s + 2.00·6-s + 4·7-s − 2.82i·8-s + 0.999·9-s − 1.41i·10-s + (3 − 1.41i)11-s + 2.82i·12-s + 4.24i·13-s + 5.65i·14-s + 1.41i·15-s + 4.00·16-s − 1.41i·17-s + ⋯
L(s)  = 1  + 0.999i·2-s − 0.816i·3-s − 1.00·4-s − 0.447·5-s + 0.816·6-s + 1.51·7-s − 1.00i·8-s + 0.333·9-s − 0.447i·10-s + (0.904 − 0.426i)11-s + 0.816i·12-s + 1.17i·13-s + 1.51i·14-s + 0.365i·15-s + 1.00·16-s − 0.342i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 - 0.426i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.904 - 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $0.904 - 0.426i$
Analytic conductor: \(1.75670\)
Root analytic conductor: \(1.32540\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{220} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 220,\ (\ :1/2),\ 0.904 - 0.426i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21033 + 0.270979i\)
\(L(\frac12)\) \(\approx\) \(1.21033 + 0.270979i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41iT \)
5 \( 1 + T \)
11 \( 1 + (-3 + 1.41i)T \)
good3 \( 1 + 1.41iT - 3T^{2} \)
7 \( 1 - 4T + 7T^{2} \)
13 \( 1 - 4.24iT - 13T^{2} \)
17 \( 1 + 1.41iT - 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 + 7.07iT - 23T^{2} \)
29 \( 1 - 2.82iT - 29T^{2} \)
31 \( 1 - 8.48iT - 31T^{2} \)
37 \( 1 + 10T + 37T^{2} \)
41 \( 1 - 2.82iT - 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + 7.07iT - 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + 2.82iT - 59T^{2} \)
61 \( 1 - 8.48iT - 61T^{2} \)
67 \( 1 - 4.24iT - 67T^{2} \)
71 \( 1 - 5.65iT - 71T^{2} \)
73 \( 1 + 4.24iT - 73T^{2} \)
79 \( 1 + 2T + 79T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−12.32572159225173888534868318232, −11.69713197133463721218562939558, −10.41513279393201236750496220607, −8.767396873938303140236528706261, −8.387923182954024130896192709445, −7.09540129258616420149287833856, −6.65964606940817689731672563890, −5.01451777206188100954596589382, −4.10392218391271091879316423678, −1.47376018620487891796086728364, 1.65409330999915474272984156560, 3.63542823351505933271264626737, 4.46632048790958852214627371168, 5.42541883623770981186067896708, 7.57401679828664294672075292699, 8.461249364944985126535333423822, 9.544158882161330291819850891124, 10.42720885594776132092103681591, 11.23053615138914854274494805200, 11.91934166331702961527607756285

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