Properties

Label 2-12e3-8.5-c1-0-14
Degree $2$
Conductor $1728$
Sign $0.965 - 0.258i$
Analytic cond. $13.7981$
Root an. cond. $3.71458$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73·7-s + 1.73i·13-s i·19-s + 5·25-s + 10.3·31-s + 12.1i·37-s − 8i·43-s − 4·49-s + 8.66i·61-s − 11i·67-s + 17·73-s + 12.1·79-s + 2.99i·91-s − 5·97-s + 19.0·103-s + ⋯
L(s)  = 1  + 0.654·7-s + 0.480i·13-s − 0.229i·19-s + 25-s + 1.86·31-s + 1.99i·37-s − 1.21i·43-s − 0.571·49-s + 1.10i·61-s − 1.34i·67-s + 1.98·73-s + 1.36·79-s + 0.314i·91-s − 0.507·97-s + 1.87·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.965 - 0.258i$
Analytic conductor: \(13.7981\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1/2),\ 0.965 - 0.258i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.894495617\)
\(L(\frac12)\) \(\approx\) \(1.894495617\)
\(L(\frac{3}{2})\) 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 \( 1 - 5T^{2} \)
7 \( 1 - 1.73T + 7T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 - 1.73iT - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 10.3T + 31T^{2} \)
37 \( 1 - 12.1iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 - 8.66iT - 61T^{2} \)
67 \( 1 + 11iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 17T + 73T^{2} \)
79 \( 1 - 12.1T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 5T + 97T^{2} \)
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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

−9.304324483432466897312238188437, −8.483503366787335107705794034378, −7.924123052188842055213415692599, −6.87320875095447504580374341809, −6.28948264343811938105833470180, −5.04779733024366273939123354785, −4.58134959694985719040420210791, −3.37972452626993171615754740985, −2.30603590690326646081458245803, −1.07809220747594361840678521613, 0.921046181872475339632389313767, 2.23853284994544826539669118115, 3.28875549578341305780087299540, 4.41084653230360678798007462069, 5.13211595768645758965032608384, 6.06037773090367456905279098070, 6.91114505886306914597025371342, 7.87221093595269911071304161404, 8.354401624463261664335910153022, 9.286921785096096016659372841441

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