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Properties

Label 2-220-44.7-c1-0-8
Degree $2$
Conductor $220$
Sign $-0.299 - 0.954i$
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  + (0.432 + 1.34i)2-s + (2.09 + 0.680i)3-s + (−1.62 + 1.16i)4-s + (−0.809 + 0.587i)5-s + (−0.0101 + 3.11i)6-s + (0.548 + 1.68i)7-s + (−2.27 − 1.68i)8-s + (1.49 + 1.08i)9-s + (−1.14 − 0.834i)10-s + (−0.218 − 3.30i)11-s + (−4.19 + 1.33i)12-s + (0.368 − 0.506i)13-s + (−2.03 + 1.46i)14-s + (−2.09 + 0.680i)15-s + (1.28 − 3.78i)16-s + (4.68 + 6.44i)17-s + ⋯
L(s)  = 1  + (0.305 + 0.952i)2-s + (1.20 + 0.392i)3-s + (−0.812 + 0.582i)4-s + (−0.361 + 0.262i)5-s + (−0.00413 + 1.27i)6-s + (0.207 + 0.638i)7-s + (−0.803 − 0.595i)8-s + (0.499 + 0.362i)9-s + (−0.360 − 0.264i)10-s + (−0.0659 − 0.997i)11-s + (−1.21 + 0.385i)12-s + (0.102 − 0.140i)13-s + (−0.544 + 0.392i)14-s + (−0.540 + 0.175i)15-s + (0.321 − 0.946i)16-s + (1.13 + 1.56i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.299 - 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.299 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $-0.299 - 0.954i$
Analytic conductor: \(1.75670\)
Root analytic conductor: \(1.32540\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{220} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 220,\ (\ :1/2),\ -0.299 - 0.954i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03471 + 1.40923i\)
\(L(\frac12)\) \(\approx\) \(1.03471 + 1.40923i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.432 - 1.34i)T \)
5 \( 1 + (0.809 - 0.587i)T \)
11 \( 1 + (0.218 + 3.30i)T \)
good3 \( 1 + (-2.09 - 0.680i)T + (2.42 + 1.76i)T^{2} \)
7 \( 1 + (-0.548 - 1.68i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-0.368 + 0.506i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-4.68 - 6.44i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.50 + 4.62i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 4.01iT - 23T^{2} \)
29 \( 1 + (0.923 - 0.300i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (2.21 - 3.05i)T + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.35 - 4.18i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (7.91 + 2.57i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 - 8.42T + 43T^{2} \)
47 \( 1 + (7.43 + 2.41i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (7.51 + 5.45i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (2.45 - 0.798i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (-5.77 - 7.95i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + 7.43iT - 67T^{2} \)
71 \( 1 + (1.04 + 1.44i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (3.83 - 1.24i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-2.58 - 1.87i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-6.14 + 4.46i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 6.28T + 89T^{2} \)
97 \( 1 + (-4.44 - 3.23i)T + (29.9 + 92.2i)T^{2} \)
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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

−12.89117515523684017696166346025, −11.81410661661471810520689903636, −10.43070376746651034797880349017, −9.167937270078684718505649841779, −8.440818045531642945773951962863, −7.901773788496898132070206740206, −6.46389027198463492098946901758, −5.31330648831270028557240673714, −3.82589693985355234562729882435, −2.98034603420950913926048711534, 1.55274375126064292882190770778, 3.01524094365817335754302515676, 4.07371838323940531584049554323, 5.34912237688308720129445841938, 7.37668480590789113062344254689, 7.996366551439850715092230136532, 9.339235515811333709269799791263, 9.843640272701536646224709124777, 11.19904516796628647829935799774, 12.13240812128749095083176541770

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