Properties

Label 2-4400-1.1-c1-0-77
Degree $2$
Conductor $4400$
Sign $-1$
Analytic cond. $35.1341$
Root an. cond. $5.92740$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 7-s − 2·9-s + 11-s + 17-s − 19-s − 21-s − 5·27-s − 29-s + 31-s + 33-s + 37-s − 6·43-s − 8·47-s − 6·49-s + 51-s + 9·53-s − 57-s − 4·59-s − 7·61-s + 2·63-s + 4·67-s − 5·71-s + 14·73-s − 77-s − 4·79-s + 81-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s − 2/3·9-s + 0.301·11-s + 0.242·17-s − 0.229·19-s − 0.218·21-s − 0.962·27-s − 0.185·29-s + 0.179·31-s + 0.174·33-s + 0.164·37-s − 0.914·43-s − 1.16·47-s − 6/7·49-s + 0.140·51-s + 1.23·53-s − 0.132·57-s − 0.520·59-s − 0.896·61-s + 0.251·63-s + 0.488·67-s − 0.593·71-s + 1.63·73-s − 0.113·77-s − 0.450·79-s + 1/9·81-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(4400\)    =    \(2^{4} \cdot 5^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(35.1341\)
Root analytic conductor: \(5.92740\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{4400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4400,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
7 \( 1 + T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 5 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 7 T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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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

−8.227326837622866265008796590544, −7.30585158597649557226756040516, −6.55580953383084523336225101016, −5.84613831007330099419015653788, −5.04523007135713372903963223204, −4.04919364705774213673393910855, −3.28418393568304306414564053333, −2.58966013719884848890127061396, −1.51155837193770356965403922878, 0, 1.51155837193770356965403922878, 2.58966013719884848890127061396, 3.28418393568304306414564053333, 4.04919364705774213673393910855, 5.04523007135713372903963223204, 5.84613831007330099419015653788, 6.55580953383084523336225101016, 7.30585158597649557226756040516, 8.227326837622866265008796590544

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