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Properties

Label 2-12e3-8.5-c1-0-12
Degree $2$
Conductor $1728$
Sign $0.707 - 0.707i$
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  + 5.19·7-s + 5.19i·13-s + 7i·19-s + 5·25-s − 10.3·31-s − 5.19i·37-s + 8i·43-s + 20·49-s − 15.5i·61-s + 5i·67-s − 7·73-s − 5.19·79-s + 27i·91-s + 19·97-s + 15.5·103-s + ⋯
L(s)  = 1  + 1.96·7-s + 1.44i·13-s + 1.60i·19-s + 25-s − 1.86·31-s − 0.854i·37-s + 1.21i·43-s + 2.85·49-s − 1.99i·61-s + 0.610i·67-s − 0.819·73-s − 0.584·79-s + 2.83i·91-s + 1.92·97-s + 1.53·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.707 - 0.707i$
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: \((2,\ 1728,\ (\ :1/2),\ 0.707 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.126499430\)
\(L(\frac12)\) \(\approx\) \(2.126499430\)
\(L(\frac{3}{2})\) 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 \( 1 - 5.19T + 7T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 - 5.19iT - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 7iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 10.3T + 31T^{2} \)
37 \( 1 + 5.19iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 8iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 + 15.5iT - 61T^{2} \)
67 \( 1 - 5iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 7T + 73T^{2} \)
79 \( 1 + 5.19T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 19T + 97T^{2} \)
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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

−9.223597894169540281995454776058, −8.637885885691099262528953318880, −7.81497807309749860405629903649, −7.25023701180873607119826579376, −6.14055620025216768310432286080, −5.22193300531168785282336725978, −4.51152190074165218393384697165, −3.69616270441936405010368922630, −2.06808796982476330780568652731, −1.46644145677579914029027627327, 0.878657737742519940725745708193, 2.07808415771153746447899347587, 3.15801042324966766812408215396, 4.46014542533290480136668642150, 5.09988964480557560625905683404, 5.72643963790198297914648659281, 7.15280139764308751821372601951, 7.56575287474365501785755560286, 8.584689249673433565612438966446, 8.858991572456678357290907282883

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