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Properties

Label 2-12e3-36.23-c1-0-15
Degree $2$
Conductor $1728$
Sign $0.642 + 0.766i$
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  + (1.5 + 0.866i)5-s + (1.5 − 0.866i)7-s + (−1.5 − 2.59i)11-s + (−2.5 + 4.33i)13-s − 6.92i·17-s − 3.46i·19-s + (4.5 − 7.79i)23-s + (−1 − 1.73i)25-s + (1.5 − 0.866i)29-s + (4.5 + 2.59i)31-s + 3·35-s − 2·37-s + (4.5 + 2.59i)41-s + (4.5 − 2.59i)43-s + (1.5 + 2.59i)47-s + ⋯
L(s)  = 1  + (0.670 + 0.387i)5-s + (0.566 − 0.327i)7-s + (−0.452 − 0.783i)11-s + (−0.693 + 1.20i)13-s − 1.68i·17-s − 0.794i·19-s + (0.938 − 1.62i)23-s + (−0.200 − 0.346i)25-s + (0.278 − 0.160i)29-s + (0.808 + 0.466i)31-s + 0.507·35-s − 0.328·37-s + (0.702 + 0.405i)41-s + (0.686 − 0.396i)43-s + (0.218 + 0.378i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.642 + 0.766i$
Analytic conductor: \(13.7981\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1/2),\ 0.642 + 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.877651245\)
\(L(\frac12)\) \(\approx\) \(1.877651245\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 - 0.866i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.5 + 0.866i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.5 - 4.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 6.92iT - 17T^{2} \)
19 \( 1 + 3.46iT - 19T^{2} \)
23 \( 1 + (-4.5 + 7.79i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.5 + 0.866i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.5 - 2.59i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + (-4.5 - 2.59i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.5 + 2.59i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.5 - 2.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (-1.5 + 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.5 - 4.33i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 + (-7.5 + 4.33i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.5 + 12.9i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 6.92iT - 89T^{2} \)
97 \( 1 + (-2.5 - 4.33i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.192132200815693085272899144355, −8.562939022361898135989428819840, −7.47086937541801507974632453170, −6.85541687127157096843633362825, −6.09601484858676204028385420480, −4.89666072598051398481400272076, −4.53321983829574038200600501389, −2.89832325629917406600288264131, −2.34278234571170922057114869899, −0.74312456824899457071961554420, 1.37656671349406079529132439502, 2.26194593262034754642432512252, 3.47606861610909727480640568158, 4.64344647824210211937101944329, 5.49703913739822491874433772359, 5.87601801070715707801943004417, 7.22578467268255709773163177507, 7.894667363123197364823077023928, 8.564409338357346597944712579120, 9.565798022651884520080652505112

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