Properties

Label 2-140e2-1.1-c1-0-21
Degree $2$
Conductor $19600$
Sign $1$
Analytic cond. $156.506$
Root an. cond. $12.5102$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 9-s − 2·17-s − 2·19-s + 8·23-s + 4·27-s + 2·29-s + 4·31-s + 6·37-s + 2·41-s + 8·43-s + 4·47-s + 4·51-s + 10·53-s + 4·57-s + 6·59-s − 4·61-s − 12·67-s − 16·69-s − 14·73-s + 8·79-s − 11·81-s − 6·83-s − 4·87-s − 10·89-s − 8·93-s − 2·97-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/3·9-s − 0.485·17-s − 0.458·19-s + 1.66·23-s + 0.769·27-s + 0.371·29-s + 0.718·31-s + 0.986·37-s + 0.312·41-s + 1.21·43-s + 0.583·47-s + 0.560·51-s + 1.37·53-s + 0.529·57-s + 0.781·59-s − 0.512·61-s − 1.46·67-s − 1.92·69-s − 1.63·73-s + 0.900·79-s − 1.22·81-s − 0.658·83-s − 0.428·87-s − 1.05·89-s − 0.829·93-s − 0.203·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(19600\)    =    \(2^{4} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(156.506\)
Root analytic conductor: \(12.5102\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{19600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 19600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.281955822\)
\(L(\frac12)\) \(\approx\) \(1.281955822\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good3 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.82405765005163, −15.09693119252884, −14.79230825543658, −14.00675929628737, −13.36900657864091, −12.91073723587234, −12.28972355809745, −11.81498146814740, −11.21878183900450, −10.80610808729591, −10.36002821240915, −9.604735318591604, −8.882095097205039, −8.518256119484362, −7.563653662121162, −7.020168058158480, −6.436078599670725, −5.843726739378018, −5.345202499043781, −4.539733454781880, −4.204446779708564, −2.999772450411071, −2.482486436164423, −1.247481123266087, −0.5635329325736111, 0.5635329325736111, 1.247481123266087, 2.482486436164423, 2.999772450411071, 4.204446779708564, 4.539733454781880, 5.345202499043781, 5.843726739378018, 6.436078599670725, 7.020168058158480, 7.563653662121162, 8.518256119484362, 8.882095097205039, 9.604735318591604, 10.36002821240915, 10.80610808729591, 11.21878183900450, 11.81498146814740, 12.28972355809745, 12.91073723587234, 13.36900657864091, 14.00675929628737, 14.79230825543658, 15.09693119252884, 15.82405765005163

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