Properties

Label 2-140e2-1.1-c1-0-36
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  − 3·9-s + 4·11-s + 2·13-s − 6·17-s + 8·19-s + 6·29-s + 8·31-s + 2·37-s − 2·41-s − 4·43-s + 8·47-s − 6·53-s + 6·61-s − 4·67-s + 8·71-s + 10·73-s − 16·79-s + 9·81-s − 8·83-s + 6·89-s − 6·97-s − 12·99-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 9-s + 1.20·11-s + 0.554·13-s − 1.45·17-s + 1.83·19-s + 1.11·29-s + 1.43·31-s + 0.328·37-s − 0.312·41-s − 0.609·43-s + 1.16·47-s − 0.824·53-s + 0.768·61-s − 0.488·67-s + 0.949·71-s + 1.17·73-s − 1.80·79-s + 81-s − 0.878·83-s + 0.635·89-s − 0.609·97-s − 1.20·99-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-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

\(L(1)\) \(\approx\) \(2.365210331\)
\(L(\frac12)\) \(\approx\) \(2.365210331\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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

−15.73507518800326, −15.23469211060141, −14.43758337670298, −13.99064301634751, −13.70629004569801, −13.08544317734785, −12.14856124933276, −11.81677320811003, −11.36551152278630, −10.86827709856012, −10.04039764562615, −9.480341541973650, −8.903369077750355, −8.464085616273344, −7.873461672744873, −6.944121889410786, −6.535327628670816, −5.968214179511449, −5.228005326238120, −4.543853997241394, −3.856754815901654, −3.093402179602565, −2.517813570394899, −1.432160043414152, −0.6822103823196131, 0.6822103823196131, 1.432160043414152, 2.517813570394899, 3.093402179602565, 3.856754815901654, 4.543853997241394, 5.228005326238120, 5.968214179511449, 6.535327628670816, 6.944121889410786, 7.873461672744873, 8.464085616273344, 8.903369077750355, 9.480341541973650, 10.04039764562615, 10.86827709856012, 11.36551152278630, 11.81677320811003, 12.14856124933276, 13.08544317734785, 13.70629004569801, 13.99064301634751, 14.43758337670298, 15.23469211060141, 15.73507518800326

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