Properties

Label 2-12e3-1.1-c1-0-7
Degree $2$
Conductor $1728$
Sign $1$
Analytic cond. $13.7981$
Root an. cond. $3.71458$
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  − 2·5-s + 3·7-s − 6·11-s + 3·13-s − 2·17-s + 3·19-s + 6·23-s − 25-s + 8·29-s − 6·35-s − 7·37-s + 8·41-s + 12·43-s + 6·47-s + 2·49-s − 4·53-s + 12·55-s − 6·59-s + 61-s − 6·65-s + 3·67-s + 12·71-s − 15·73-s − 18·77-s + 9·79-s + 12·83-s + 4·85-s + ⋯
L(s)  = 1  − 0.894·5-s + 1.13·7-s − 1.80·11-s + 0.832·13-s − 0.485·17-s + 0.688·19-s + 1.25·23-s − 1/5·25-s + 1.48·29-s − 1.01·35-s − 1.15·37-s + 1.24·41-s + 1.82·43-s + 0.875·47-s + 2/7·49-s − 0.549·53-s + 1.61·55-s − 0.781·59-s + 0.128·61-s − 0.744·65-s + 0.366·67-s + 1.42·71-s − 1.75·73-s − 2.05·77-s + 1.01·79-s + 1.31·83-s + 0.433·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $1$
Analytic conductor: \(13.7981\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1728} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.518894481\)
\(L(\frac12)\) \(\approx\) \(1.518894481\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 - 3 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 3 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 15 T + p T^{2} \)
79 \( 1 - 9 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 9 T + p T^{2} \)
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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

−9.092965331861758202249959081592, −8.389260667460447760534211383269, −7.75343299782764246102555933457, −7.27738079891968835931324286954, −5.98907722179424364210084004193, −5.05657529199533656835469206720, −4.50764101424572461559505208448, −3.34885698116883702758597808395, −2.35764661264624271155228051702, −0.861270098074716828308690272571, 0.861270098074716828308690272571, 2.35764661264624271155228051702, 3.34885698116883702758597808395, 4.50764101424572461559505208448, 5.05657529199533656835469206720, 5.98907722179424364210084004193, 7.27738079891968835931324286954, 7.75343299782764246102555933457, 8.389260667460447760534211383269, 9.092965331861758202249959081592

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