Properties

Label 2-12e3-1.1-c1-0-6
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 + 7-s − 2·11-s − 13-s + 6·17-s + 5·19-s − 6·23-s − 25-s − 8·29-s + 8·31-s − 2·35-s + 5·37-s + 8·41-s + 4·43-s + 10·47-s − 6·49-s − 4·53-s + 4·55-s + 14·59-s − 3·61-s + 2·65-s + 13·67-s + 4·71-s + 9·73-s − 2·77-s + 11·79-s − 12·83-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.377·7-s − 0.603·11-s − 0.277·13-s + 1.45·17-s + 1.14·19-s − 1.25·23-s − 1/5·25-s − 1.48·29-s + 1.43·31-s − 0.338·35-s + 0.821·37-s + 1.24·41-s + 0.609·43-s + 1.45·47-s − 6/7·49-s − 0.549·53-s + 0.539·55-s + 1.82·59-s − 0.384·61-s + 0.248·65-s + 1.58·67-s + 0.474·71-s + 1.05·73-s − 0.227·77-s + 1.23·79-s − 1.31·83-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

\(L(1)\) \(\approx\) \(1.417891343\)
\(L(\frac12)\) \(\approx\) \(1.417891343\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 + 3 T + p T^{2} \)
67 \( 1 - 13 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 9 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−9.517026791789933025951612270668, −8.141441334825677566937789349991, −7.86749870783756385045860969127, −7.24764130899628727569423119197, −5.92718935464768987920858473492, −5.29554848587607308034135507932, −4.23592786371004028357777676518, −3.47453539619149025125996376504, −2.35447010108907921399113958644, −0.826140575077611309933401896495, 0.826140575077611309933401896495, 2.35447010108907921399113958644, 3.47453539619149025125996376504, 4.23592786371004028357777676518, 5.29554848587607308034135507932, 5.92718935464768987920858473492, 7.24764130899628727569423119197, 7.86749870783756385045860969127, 8.141441334825677566937789349991, 9.517026791789933025951612270668

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