Properties

Label 2-12e3-1.1-c1-0-3
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  − 5-s − 3·7-s + 3·11-s − 4·17-s + 6·19-s + 6·23-s − 4·25-s − 2·29-s − 9·31-s + 3·35-s + 2·37-s + 10·41-s + 6·43-s + 6·47-s + 2·49-s + 13·53-s − 3·55-s + 12·59-s − 8·61-s + 6·67-s + 12·71-s + 9·73-s − 9·77-s + 3·83-s + 4·85-s − 14·89-s − 6·95-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.13·7-s + 0.904·11-s − 0.970·17-s + 1.37·19-s + 1.25·23-s − 4/5·25-s − 0.371·29-s − 1.61·31-s + 0.507·35-s + 0.328·37-s + 1.56·41-s + 0.914·43-s + 0.875·47-s + 2/7·49-s + 1.78·53-s − 0.404·55-s + 1.56·59-s − 1.02·61-s + 0.733·67-s + 1.42·71-s + 1.05·73-s − 1.02·77-s + 0.329·83-s + 0.433·85-s − 1.48·89-s − 0.615·95-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

\(L(1)\) \(\approx\) \(1.342039504\)
\(L(\frac12)\) \(\approx\) \(1.342039504\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 9 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 13 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 6 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 9 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−9.294658633853491510374780184936, −8.784687360409477775570593433890, −7.47224706156871722823044931973, −7.07642279360820371836072015909, −6.14570823384352111267011079375, −5.33767563665680457913764465863, −4.07336899816643905031985228211, −3.52624329017255126054742371711, −2.39632866530783962583020681104, −0.794899488559166432876984150469, 0.794899488559166432876984150469, 2.39632866530783962583020681104, 3.52624329017255126054742371711, 4.07336899816643905031985228211, 5.33767563665680457913764465863, 6.14570823384352111267011079375, 7.07642279360820371836072015909, 7.47224706156871722823044931973, 8.784687360409477775570593433890, 9.294658633853491510374780184936

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