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Properties

Label 2-4400-5.4-c1-0-17
Degree $2$
Conductor $4400$
Sign $0.447 - 0.894i$
Analytic cond. $35.1341$
Root an. cond. $5.92740$
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  − 2i·3-s + 4i·7-s − 9-s + 11-s + 4i·13-s − 4·19-s + 8·21-s − 6i·23-s − 4i·27-s + 6·29-s − 8·31-s − 2i·33-s + 2i·37-s + 8·39-s + 6·41-s + ⋯
L(s)  = 1  − 1.15i·3-s + 1.51i·7-s − 0.333·9-s + 0.301·11-s + 1.10i·13-s − 0.917·19-s + 1.74·21-s − 1.25i·23-s − 0.769i·27-s + 1.11·29-s − 1.43·31-s − 0.348i·33-s + 0.328i·37-s + 1.28·39-s + 0.937·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4400\)    =    \(2^{4} \cdot 5^{2} \cdot 11\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(35.1341\)
Root analytic conductor: \(5.92740\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4400} (4049, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4400,\ (\ :1/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.395552323\)
\(L(\frac12)\) \(\approx\) \(1.395552323\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
11 \( 1 - T \)
good3 \( 1 + 2iT - 3T^{2} \)
7 \( 1 - 4iT - 7T^{2} \)
13 \( 1 - 4iT - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 8iT - 43T^{2} \)
47 \( 1 + 6iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 - 10iT - 67T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 - 16iT - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 - 14iT - 97T^{2} \)
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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

−8.484868423424914574257910180988, −7.81789478883457532979644268571, −6.79029853021330597213138029175, −6.48592897144881120962580787079, −5.80367757595412118719316053403, −4.81270531717960688091592799991, −4.00343639222457504145259479806, −2.59012011811437673221877864792, −2.20733581041761689761448256324, −1.21091663369361519384269238451, 0.40377772005865333725877731337, 1.65155429115363665808677536462, 3.16827334640832864537205762487, 3.74676459068602669907655443151, 4.36108758406222173207937981646, 5.08452512656498444500279709867, 5.91397632472205077913348983630, 6.88503091097514978187677130889, 7.53299544089796342107674187294, 8.185068036558618712086829716703

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