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Properties

Label 2-12e3-9.7-c1-0-16
Degree $2$
Conductor $1728$
Sign $0.254 + 0.967i$
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  + (0.5 + 0.866i)5-s + (0.724 − 1.25i)7-s + (1.72 − 2.98i)11-s + (−1.94 − 3.37i)13-s + 4.89·17-s − 4·19-s + (−0.275 − 0.476i)23-s + (2 − 3.46i)25-s + (−4.94 + 8.57i)29-s + (−3.72 − 6.45i)31-s + 1.44·35-s − 8.89·37-s + (−1.05 − 1.81i)41-s + (6.17 − 10.6i)43-s + (4.17 − 7.22i)47-s + ⋯
L(s)  = 1  + (0.223 + 0.387i)5-s + (0.273 − 0.474i)7-s + (0.520 − 0.900i)11-s + (−0.540 − 0.936i)13-s + 1.18·17-s − 0.917·19-s + (−0.0573 − 0.0994i)23-s + (0.400 − 0.692i)25-s + (−0.919 + 1.59i)29-s + (−0.668 − 1.15i)31-s + 0.245·35-s − 1.46·37-s + (−0.164 − 0.284i)41-s + (0.941 − 1.63i)43-s + (0.608 − 1.05i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.254 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.254 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.254 + 0.967i$
Analytic conductor: \(13.7981\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (1153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1/2),\ 0.254 + 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.606621005\)
\(L(\frac12)\) \(\approx\) \(1.606621005\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.724 + 1.25i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.72 + 2.98i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.94 + 3.37i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 4.89T + 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + (0.275 + 0.476i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.94 - 8.57i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.72 + 6.45i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 8.89T + 37T^{2} \)
41 \( 1 + (1.05 + 1.81i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6.17 + 10.6i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.17 + 7.22i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 0.898T + 53T^{2} \)
59 \( 1 + (-0.174 - 0.301i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.949 + 1.64i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.17 + 2.03i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 11.7T + 71T^{2} \)
73 \( 1 - 4.89T + 73T^{2} \)
79 \( 1 + (4.27 - 7.40i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.72 + 4.71i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 3.10T + 89T^{2} \)
97 \( 1 + (2.94 - 5.10i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.071698686476973411864869919710, −8.399259939674251105497622727195, −7.50052460581122402232893906694, −6.87574856633566116415741715499, −5.80388060812864259137187820590, −5.26201605192546741285147422375, −3.94748068668391173135083523699, −3.26337406189175108265818819677, −2.04648328026271667662016722156, −0.61748662300169347866639523117, 1.46612916204850242837386191148, 2.30264260895303470375537844893, 3.68888130495642131227991742739, 4.60822594912482545536674037573, 5.34196964038995191305893000353, 6.26672527602408259595246852950, 7.15176822904438395207155597464, 7.86806182505652609854801021636, 8.893653228581055860486007262859, 9.389839703708963496595635096562

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