Properties

Label 2-4400-5.4-c1-0-5
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·7-s + 3·9-s + 11-s − 8·19-s + 8i·23-s − 10·29-s − 8·31-s + 10i·37-s − 2·41-s + 6i·43-s − 8i·47-s + 3·49-s + 14i·53-s − 4·59-s + 10·61-s + ⋯
L(s)  = 1  − 0.755i·7-s + 9-s + 0.301·11-s − 1.83·19-s + 1.66i·23-s − 1.85·29-s − 1.43·31-s + 1.64i·37-s − 0.312·41-s + 0.914i·43-s − 1.16i·47-s + 0.428·49-s + 1.92i·53-s − 0.520·59-s + 1.28·61-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

\(L(1)\) \(\approx\) \(0.8620551940\)
\(L(\frac12)\) \(\approx\) \(0.8620551940\)
\(L(\frac{3}{2})\) 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 \( 1 - 3T^{2} \)
7 \( 1 + 2iT - 7T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 8T + 19T^{2} \)
23 \( 1 - 8iT - 23T^{2} \)
29 \( 1 + 10T + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 - 10iT - 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 6iT - 43T^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 - 14iT - 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 8iT - 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + 10iT - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 - 10iT - 97T^{2} \)
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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

−8.643537153037795994690228853748, −7.60653758125453381460823461758, −7.29753960001246101262495865526, −6.50012029258434021074980432500, −5.70844906895903716326390942239, −4.74486459160823184004287558951, −3.99192640734493392743961311446, −3.49575347005544356926014876528, −2.03512810024324411219103687305, −1.32205053360290164875469187472, 0.22942666032940754763124551861, 1.86237394473426923529474242763, 2.32129383064925947799850511398, 3.76062417333265384353872895755, 4.20353733430988535345696739498, 5.20787547662399056070055629642, 5.93741964595754633704130381082, 6.73015654793833406618220027928, 7.28353569542483018708836282952, 8.233534680830904802576332703701

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