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Properties

Label 2-4410-105.104-c1-0-69
Degree $2$
Conductor $4410$
Sign $-0.0495 + 0.998i$
Analytic cond. $35.2140$
Root an. cond. $5.93414$
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  + 2-s + 4-s + (0.426 + 2.19i)5-s + 8-s + (0.426 + 2.19i)10-s − 1.28i·11-s − 6.14·13-s + 16-s − 6.52i·17-s − 6.03i·19-s + (0.426 + 2.19i)20-s − 1.28i·22-s + 2.87·23-s + (−4.63 + 1.87i)25-s − 6.14·26-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + (0.190 + 0.981i)5-s + 0.353·8-s + (0.134 + 0.694i)10-s − 0.386i·11-s − 1.70·13-s + 0.250·16-s − 1.58i·17-s − 1.38i·19-s + (0.0953 + 0.490i)20-s − 0.273i·22-s + 0.600·23-s + (−0.927 + 0.374i)25-s − 1.20·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0495 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0495 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4410\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.0495 + 0.998i$
Analytic conductor: \(35.2140\)
Root analytic conductor: \(5.93414\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4410} (4409, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4410,\ (\ :1/2),\ -0.0495 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.771862367\)
\(L(\frac12)\) \(\approx\) \(1.771862367\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
5 \( 1 + (-0.426 - 2.19i)T \)
7 \( 1 \)
good11 \( 1 + 1.28iT - 11T^{2} \)
13 \( 1 + 6.14T + 13T^{2} \)
17 \( 1 + 6.52iT - 17T^{2} \)
19 \( 1 + 6.03iT - 19T^{2} \)
23 \( 1 - 2.87T + 23T^{2} \)
29 \( 1 + 1.35iT - 29T^{2} \)
31 \( 1 + 8.65iT - 31T^{2} \)
37 \( 1 - 9.53iT - 37T^{2} \)
41 \( 1 + 8.71T + 41T^{2} \)
43 \( 1 + 5.35iT - 43T^{2} \)
47 \( 1 + 0.806iT - 47T^{2} \)
53 \( 1 + 6.66T + 53T^{2} \)
59 \( 1 + 1.59T + 59T^{2} \)
61 \( 1 - 6.35iT - 61T^{2} \)
67 \( 1 + 5.39iT - 67T^{2} \)
71 \( 1 + 15.6iT - 71T^{2} \)
73 \( 1 + 12.4T + 73T^{2} \)
79 \( 1 - 11.1T + 79T^{2} \)
83 \( 1 + 3.74iT - 83T^{2} \)
89 \( 1 - 3.63T + 89T^{2} \)
97 \( 1 - 8.76T + 97T^{2} \)
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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

−7.81337917607863968983569724565, −7.25513693337781479356441456298, −6.78040415232287952875231526686, −6.00164543226331493570029445411, −4.94412426553970291223403210701, −4.72956784277107852240434724449, −3.33822777426052440374378479034, −2.79031292094490937753794047696, −2.13703759946383614284090011154, −0.34645533647268083009451478549, 1.43518746140469687017101227214, 2.11151900025192622985433464253, 3.30929801287201278385773095785, 4.15411651504074644170660217027, 4.88416755083664332496099146993, 5.41949413859771261552206420514, 6.19493228206825140607015074330, 7.05704781133405571829790256721, 7.79323973017378997333407354290, 8.447576939698830579748155334878

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