Properties

Label 2-42e2-1.1-c3-0-48
Degree $2$
Conductor $1764$
Sign $-1$
Analytic cond. $104.079$
Root an. cond. $10.2019$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 20·5-s − 44·11-s − 44·13-s − 72·17-s + 100·19-s + 120·23-s + 275·25-s − 218·29-s − 280·31-s − 30·37-s − 120·41-s + 220·43-s − 88·47-s − 110·53-s − 880·55-s − 580·59-s + 380·61-s − 880·65-s − 980·67-s + 112·71-s − 640·73-s − 488·79-s − 660·83-s − 1.44e3·85-s − 320·89-s + 2.00e3·95-s + 248·97-s + ⋯
L(s)  = 1  + 1.78·5-s − 1.20·11-s − 0.938·13-s − 1.02·17-s + 1.20·19-s + 1.08·23-s + 11/5·25-s − 1.39·29-s − 1.62·31-s − 0.133·37-s − 0.457·41-s + 0.780·43-s − 0.273·47-s − 0.285·53-s − 2.15·55-s − 1.27·59-s + 0.797·61-s − 1.67·65-s − 1.78·67-s + 0.187·71-s − 1.02·73-s − 0.694·79-s − 0.872·83-s − 1.83·85-s − 0.381·89-s + 2.15·95-s + 0.259·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(104.079\)
Root analytic conductor: \(10.2019\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1764} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1764,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 - 4 p T + p^{3} T^{2} \)
11 \( 1 + 4 p T + p^{3} T^{2} \)
13 \( 1 + 44 T + p^{3} T^{2} \)
17 \( 1 + 72 T + p^{3} T^{2} \)
19 \( 1 - 100 T + p^{3} T^{2} \)
23 \( 1 - 120 T + p^{3} T^{2} \)
29 \( 1 + 218 T + p^{3} T^{2} \)
31 \( 1 + 280 T + p^{3} T^{2} \)
37 \( 1 + 30 T + p^{3} T^{2} \)
41 \( 1 + 120 T + p^{3} T^{2} \)
43 \( 1 - 220 T + p^{3} T^{2} \)
47 \( 1 + 88 T + p^{3} T^{2} \)
53 \( 1 + 110 T + p^{3} T^{2} \)
59 \( 1 + 580 T + p^{3} T^{2} \)
61 \( 1 - 380 T + p^{3} T^{2} \)
67 \( 1 + 980 T + p^{3} T^{2} \)
71 \( 1 - 112 T + p^{3} T^{2} \)
73 \( 1 + 640 T + p^{3} T^{2} \)
79 \( 1 + 488 T + p^{3} T^{2} \)
83 \( 1 + 660 T + p^{3} T^{2} \)
89 \( 1 + 320 T + p^{3} T^{2} \)
97 \( 1 - 248 T + p^{3} T^{2} \)
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

−8.876773211852689029295496327903, −7.51931417997634608047466272975, −7.05536129368666403548666751415, −5.89614190493857664150055268964, −5.37665403179725781443554460367, −4.73468697094938213078097199805, −3.12685146282318003479667885586, −2.35795000589208637979510405165, −1.54423732694106313288872672972, 0, 1.54423732694106313288872672972, 2.35795000589208637979510405165, 3.12685146282318003479667885586, 4.73468697094938213078097199805, 5.37665403179725781443554460367, 5.89614190493857664150055268964, 7.05536129368666403548666751415, 7.51931417997634608047466272975, 8.876773211852689029295496327903

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