Properties

Label 2-12e3-1.1-c1-0-23
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7-s − 5·13-s + 7·19-s − 5·25-s − 4·31-s − 11·37-s − 8·43-s − 6·49-s + 61-s − 5·67-s − 7·73-s + 17·79-s + 5·91-s − 19·97-s − 13·103-s − 2·109-s + ⋯
L(s)  = 1  − 0.377·7-s − 1.38·13-s + 1.60·19-s − 25-s − 0.718·31-s − 1.80·37-s − 1.21·43-s − 6/7·49-s + 0.128·61-s − 0.610·67-s − 0.819·73-s + 1.91·79-s + 0.524·91-s − 1.92·97-s − 1.28·103-s − 0.191·109-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: \(1\)
Selberg data: \((2,\ 1728,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 - 17 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 19 T + p T^{2} \)
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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

−9.093756787868480250299391378087, −8.013308166903493545899536406476, −7.33966183117961392379729992729, −6.66054107823412239247003697818, −5.50710978221504476175964668301, −4.97480875199317599127464128318, −3.74735768991078116659064649628, −2.90681031253330114231046316229, −1.71126395023482908495018202224, 0, 1.71126395023482908495018202224, 2.90681031253330114231046316229, 3.74735768991078116659064649628, 4.97480875199317599127464128318, 5.50710978221504476175964668301, 6.66054107823412239247003697818, 7.33966183117961392379729992729, 8.013308166903493545899536406476, 9.093756787868480250299391378087

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