Properties

Label 2-4400-5.4-c1-0-7
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  − 1.56i·3-s − 3.56i·7-s + 0.561·9-s − 11-s + 3.12i·13-s + 5.56i·17-s + 2.43·19-s − 5.56·21-s + 7.12i·23-s − 5.56i·27-s + 0.438·29-s − 8.68·31-s + 1.56i·33-s + 9.80i·37-s + 4.87·39-s + ⋯
L(s)  = 1  − 0.901i·3-s − 1.34i·7-s + 0.187·9-s − 0.301·11-s + 0.866i·13-s + 1.34i·17-s + 0.559·19-s − 1.21·21-s + 1.48i·23-s − 1.07i·27-s + 0.0814·29-s − 1.55·31-s + 0.271i·33-s + 1.61i·37-s + 0.780·39-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

\(L(1)\) \(\approx\) \(0.9746130019\)
\(L(\frac12)\) \(\approx\) \(0.9746130019\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.56iT - 3T^{2} \)
7 \( 1 + 3.56iT - 7T^{2} \)
13 \( 1 - 3.12iT - 13T^{2} \)
17 \( 1 - 5.56iT - 17T^{2} \)
19 \( 1 - 2.43T + 19T^{2} \)
23 \( 1 - 7.12iT - 23T^{2} \)
29 \( 1 - 0.438T + 29T^{2} \)
31 \( 1 + 8.68T + 31T^{2} \)
37 \( 1 - 9.80iT - 37T^{2} \)
41 \( 1 + 10T + 41T^{2} \)
43 \( 1 - 5.12iT - 43T^{2} \)
47 \( 1 - 7.12iT - 47T^{2} \)
53 \( 1 - 4.43iT - 53T^{2} \)
59 \( 1 + 13.3T + 59T^{2} \)
61 \( 1 + 3.56T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 2.43T + 71T^{2} \)
73 \( 1 + 4.87iT - 73T^{2} \)
79 \( 1 - 0.876T + 79T^{2} \)
83 \( 1 - 10iT - 83T^{2} \)
89 \( 1 + 9.80T + 89T^{2} \)
97 \( 1 - 17.1iT - 97T^{2} \)
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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

−8.110261840662203679026480679067, −7.74620461221440013431679345895, −7.02967065300111536687058032824, −6.56868931874860163005311744062, −5.69738749703018315766998581253, −4.64491323993515779165896143329, −3.94108655758114082327860326115, −3.15075563995483384195234649414, −1.61761709454265143219183247712, −1.38612537383633955218579482400, 0.26409907189655660251700045671, 1.97442570622843344303676868748, 2.86003278064770651827190945593, 3.56717497397309925695377811318, 4.64555391704570193856941629694, 5.30486356485303228122782835710, 5.66890499590015582217116237835, 6.84194054942055599897199888675, 7.49847890989468551586155745252, 8.462918012541182543063096125326

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