Properties

Label 2-440-1.1-c1-0-7
Degree $2$
Conductor $440$
Sign $1$
Analytic cond. $3.51341$
Root an. cond. $1.87441$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 5-s + 7-s + 6·9-s − 11-s − 6·13-s + 3·15-s + 3·17-s − 5·19-s + 3·21-s − 2·23-s + 25-s + 9·27-s − 5·29-s + 5·31-s − 3·33-s + 35-s − 37-s − 18·39-s − 2·41-s + 12·43-s + 6·45-s − 2·47-s − 6·49-s + 9·51-s − 13·53-s − 55-s + ⋯
L(s)  = 1  + 1.73·3-s + 0.447·5-s + 0.377·7-s + 2·9-s − 0.301·11-s − 1.66·13-s + 0.774·15-s + 0.727·17-s − 1.14·19-s + 0.654·21-s − 0.417·23-s + 1/5·25-s + 1.73·27-s − 0.928·29-s + 0.898·31-s − 0.522·33-s + 0.169·35-s − 0.164·37-s − 2.88·39-s − 0.312·41-s + 1.82·43-s + 0.894·45-s − 0.291·47-s − 6/7·49-s + 1.26·51-s − 1.78·53-s − 0.134·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(440\)    =    \(2^{3} \cdot 5 \cdot 11\)
Sign: $1$
Analytic conductor: \(3.51341\)
Root analytic conductor: \(1.87441\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 440,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.491010532\)
\(L(\frac12)\) \(\approx\) \(2.491010532\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
11 \( 1 + T \)
good3 \( 1 - p T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 13 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 - 15 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
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

−10.89766511338015233329150568707, −9.786029738762734281399683357155, −9.483006760033315076271652715999, −8.241837378038505346249898701453, −7.79929638968846771610897877135, −6.73950043879569860448501959504, −5.18236780553163224027614577017, −4.07821246743313595622649858247, −2.77079934605572561635474291231, −1.97350758509779452319901265349, 1.97350758509779452319901265349, 2.77079934605572561635474291231, 4.07821246743313595622649858247, 5.18236780553163224027614577017, 6.73950043879569860448501959504, 7.79929638968846771610897877135, 8.241837378038505346249898701453, 9.483006760033315076271652715999, 9.786029738762734281399683357155, 10.89766511338015233329150568707

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