Properties

Label 2-12e3-12.11-c1-0-7
Degree $2$
Conductor $1728$
Sign $-i$
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  + 0.317i·5-s + 1.44i·7-s + 1.09·11-s − 2.89·13-s + 3.46i·17-s − 4.89i·19-s + 2.82·23-s + 4.89·25-s + 9.12i·29-s + 7.44i·31-s − 0.460·35-s − 4.89·37-s + 9.75i·41-s − 6.89i·43-s − 9.12·47-s + ⋯
L(s)  = 1  + 0.142i·5-s + 0.547i·7-s + 0.330·11-s − 0.804·13-s + 0.840i·17-s − 1.12i·19-s + 0.589·23-s + 0.979·25-s + 1.69i·29-s + 1.33i·31-s − 0.0778·35-s − 0.805·37-s + 1.52i·41-s − 1.05i·43-s − 1.33·47-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-i$
Analytic conductor: \(13.7981\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (1727, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.363479697\)
\(L(\frac12)\) \(\approx\) \(1.363479697\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.317iT - 5T^{2} \)
7 \( 1 - 1.44iT - 7T^{2} \)
11 \( 1 - 1.09T + 11T^{2} \)
13 \( 1 + 2.89T + 13T^{2} \)
17 \( 1 - 3.46iT - 17T^{2} \)
19 \( 1 + 4.89iT - 19T^{2} \)
23 \( 1 - 2.82T + 23T^{2} \)
29 \( 1 - 9.12iT - 29T^{2} \)
31 \( 1 - 7.44iT - 31T^{2} \)
37 \( 1 + 4.89T + 37T^{2} \)
41 \( 1 - 9.75iT - 41T^{2} \)
43 \( 1 + 6.89iT - 43T^{2} \)
47 \( 1 + 9.12T + 47T^{2} \)
53 \( 1 - 4.41iT - 53T^{2} \)
59 \( 1 + 9.12T + 59T^{2} \)
61 \( 1 + 4T + 61T^{2} \)
67 \( 1 - 5.10iT - 67T^{2} \)
71 \( 1 - 7.56T + 71T^{2} \)
73 \( 1 - 1.89T + 73T^{2} \)
79 \( 1 + 2iT - 79T^{2} \)
83 \( 1 - 15.8T + 83T^{2} \)
89 \( 1 - 11.9iT - 89T^{2} \)
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

−9.227962935585059355535013677923, −8.940254043545858080850180668166, −7.998767567836365882664195185164, −6.92263869672202396305037888579, −6.56431280351604046835907917137, −5.23937428807383987291611273361, −4.81467098180068386576103182849, −3.45398995811420679498994638287, −2.65466276443996398665011776040, −1.37706470852213439792547562588, 0.53299476197290196490512769634, 1.96958759958764648889852690180, 3.13933296115095767517559616887, 4.16835175865733896843310382648, 4.93242004473959737927264825101, 5.90083940782119082330652527223, 6.81665015127744345983934365911, 7.55162682111350830522495621778, 8.226889396218829029113350494867, 9.303559282348069994017498776353

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