Properties

Label 2-12e3-1.1-c1-0-4
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  − 5·7-s + 7·13-s − 19-s − 5·25-s + 4·31-s + 37-s + 8·43-s + 18·49-s + 13·61-s + 11·67-s + 17·73-s + 13·79-s − 35·91-s + 5·97-s + 7·103-s − 2·109-s + ⋯
L(s)  = 1  − 1.88·7-s + 1.94·13-s − 0.229·19-s − 25-s + 0.718·31-s + 0.164·37-s + 1.21·43-s + 18/7·49-s + 1.66·61-s + 1.34·67-s + 1.98·73-s + 1.46·79-s − 3.66·91-s + 0.507·97-s + 0.689·103-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: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.363034090\)
\(L(\frac12)\) \(\approx\) \(1.363034090\)
\(L(\frac{3}{2})\) 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 \( 1 + 5 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 7 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
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 \( 1 - 13 T + p T^{2} \)
67 \( 1 - 11 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 17 T + p T^{2} \)
79 \( 1 - 13 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 5 T + p T^{2} \)
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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

−9.353701574803802413804076848425, −8.650409645340224009044341041842, −7.78450025598181104413322457077, −6.59048869509056337119790477940, −6.31866267091677471011901067110, −5.48216093066766077645578315510, −3.94546281439316433923646778556, −3.55340608613286864344461522402, −2.42247921633551671696045563684, −0.798653294514055210127523925209, 0.798653294514055210127523925209, 2.42247921633551671696045563684, 3.55340608613286864344461522402, 3.94546281439316433923646778556, 5.48216093066766077645578315510, 6.31866267091677471011901067110, 6.59048869509056337119790477940, 7.78450025598181104413322457077, 8.650409645340224009044341041842, 9.353701574803802413804076848425

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