Properties

Label 2-448-1.1-c1-0-4
Degree $2$
Conductor $448$
Sign $1$
Analytic cond. $3.57729$
Root an. cond. $1.89137$
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  + 2·3-s + 7-s + 9-s + 4·13-s + 6·17-s − 2·19-s + 2·21-s − 5·25-s − 4·27-s + 6·29-s − 4·31-s − 2·37-s + 8·39-s + 6·41-s − 8·43-s − 12·47-s + 49-s + 12·51-s − 6·53-s − 4·57-s + 6·59-s − 8·61-s + 63-s + 4·67-s + 2·73-s − 10·75-s + 8·79-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.377·7-s + 1/3·9-s + 1.10·13-s + 1.45·17-s − 0.458·19-s + 0.436·21-s − 25-s − 0.769·27-s + 1.11·29-s − 0.718·31-s − 0.328·37-s + 1.28·39-s + 0.937·41-s − 1.21·43-s − 1.75·47-s + 1/7·49-s + 1.68·51-s − 0.824·53-s − 0.529·57-s + 0.781·59-s − 1.02·61-s + 0.125·63-s + 0.488·67-s + 0.234·73-s − 1.15·75-s + 0.900·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $1$
Analytic conductor: \(3.57729\)
Root analytic conductor: \(1.89137\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{448} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.101530499\)
\(L(\frac12)\) \(\approx\) \(2.101530499\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

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

−11.06714116909225412617528172908, −10.04491938039307927961472194001, −9.195409247501138715663586603039, −8.222985047096640302551257734717, −7.87021772021814993834479658639, −6.47781377385923123584313505983, −5.36663802373794330843690451326, −3.93522119575008455702519349301, −3.07302685938302839530976846531, −1.64950502134773404742708641242, 1.64950502134773404742708641242, 3.07302685938302839530976846531, 3.93522119575008455702519349301, 5.36663802373794330843690451326, 6.47781377385923123584313505983, 7.87021772021814993834479658639, 8.222985047096640302551257734717, 9.195409247501138715663586603039, 10.04491938039307927961472194001, 11.06714116909225412617528172908

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