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Properties

Label 2-220-20.3-c1-0-14
Degree $2$
Conductor $220$
Sign $-0.850 + 0.525i$
Analytic cond. $1.75670$
Root an. cond. $1.32540$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − i)2-s + (−2 + 2i)3-s + 2i·4-s + (−1 + 2i)5-s + 4·6-s + (−1 − i)7-s + (2 − 2i)8-s − 5i·9-s + (3 − i)10-s + i·11-s + (−4 − 4i)12-s + (−4 − 4i)13-s + 2i·14-s + (−2 − 6i)15-s − 4·16-s + (2 − 2i)17-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + (−1.15 + 1.15i)3-s + i·4-s + (−0.447 + 0.894i)5-s + 1.63·6-s + (−0.377 − 0.377i)7-s + (0.707 − 0.707i)8-s − 1.66i·9-s + (0.948 − 0.316i)10-s + 0.301i·11-s + (−1.15 − 1.15i)12-s + (−1.10 − 1.10i)13-s + 0.534i·14-s + (−0.516 − 1.54i)15-s − 16-s + (0.485 − 0.485i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $-0.850 + 0.525i$
Analytic conductor: \(1.75670\)
Root analytic conductor: \(1.32540\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{220} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 220,\ (\ :1/2),\ -0.850 + 0.525i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + i)T \)
5 \( 1 + (1 - 2i)T \)
11 \( 1 - iT \)
good3 \( 1 + (2 - 2i)T - 3iT^{2} \)
7 \( 1 + (1 + i)T + 7iT^{2} \)
13 \( 1 + (4 + 4i)T + 13iT^{2} \)
17 \( 1 + (-2 + 2i)T - 17iT^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 + (4 - 4i)T - 23iT^{2} \)
29 \( 1 - 2iT - 29T^{2} \)
31 \( 1 + 8iT - 31T^{2} \)
37 \( 1 + (5 - 5i)T - 37iT^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + (-1 + i)T - 43iT^{2} \)
47 \( 1 + (4 + 4i)T + 47iT^{2} \)
53 \( 1 + (-9 - 9i)T + 53iT^{2} \)
59 \( 1 + 8T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 + 67iT^{2} \)
71 \( 1 - 12iT - 71T^{2} \)
73 \( 1 + (-2 - 2i)T + 73iT^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 + (-1 + i)T - 83iT^{2} \)
89 \( 1 - 6iT - 89T^{2} \)
97 \( 1 + (3 - 3i)T - 97iT^{2} \)
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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

−11.74232555403500297799651086096, −10.71756217274233987093872303941, −10.06476964818689916668247879107, −9.656433953158384843371413337775, −7.85972654731057360047242333339, −6.95985893794383838459495489488, −5.44283590408288886221109332831, −4.10408770385390336147271368246, −3.05343803685162763025309296745, 0, 1.65791358384701540528400149726, 4.77350202420320666474275227815, 5.73192741945282116023582038379, 6.67731876598189408121292457164, 7.54182331162429303070473363800, 8.526512738328720310933327079795, 9.597462312647546564069275835363, 10.82321017671691156828651678220, 12.08468361701946296351947901692

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