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Properties

Label 2-12e3-36.23-c1-0-17
Degree $2$
Conductor $1728$
Sign $0.315 + 0.948i$
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  + (0.686 + 0.396i)5-s + (2.35 − 1.35i)7-s + (−1.71 − 2.96i)11-s + (1.68 − 2.92i)13-s + 2.52i·17-s − 2.20i·19-s + (1.07 − 1.86i)23-s + (−2.18 − 3.78i)25-s + (−0.686 + 0.396i)29-s + (−1.47 − 0.852i)31-s + 2.15·35-s − 4.74·37-s + (−0.127 − 0.0737i)41-s + (6.01 − 3.47i)43-s + (−5.77 − 10.0i)47-s + ⋯
L(s)  = 1  + (0.306 + 0.177i)5-s + (0.888 − 0.513i)7-s + (−0.516 − 0.894i)11-s + (0.467 − 0.809i)13-s + 0.612i·17-s − 0.506i·19-s + (0.224 − 0.388i)23-s + (−0.437 − 0.757i)25-s + (−0.127 + 0.0735i)29-s + (−0.265 − 0.153i)31-s + 0.363·35-s − 0.780·37-s + (−0.0199 − 0.0115i)41-s + (0.917 − 0.529i)43-s + (−0.842 − 1.45i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.315 + 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.315 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.315 + 0.948i$
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: \((2,\ 1728,\ (\ :1/2),\ 0.315 + 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.801299745\)
\(L(\frac12)\) \(\approx\) \(1.801299745\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.686 - 0.396i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-2.35 + 1.35i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.71 + 2.96i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.68 + 2.92i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 2.52iT - 17T^{2} \)
19 \( 1 + 2.20iT - 19T^{2} \)
23 \( 1 + (-1.07 + 1.86i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.686 - 0.396i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.47 + 0.852i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 4.74T + 37T^{2} \)
41 \( 1 + (0.127 + 0.0737i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6.01 + 3.47i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.77 + 10.0i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 8.51iT - 53T^{2} \)
59 \( 1 + (-2.58 + 4.48i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.68 - 2.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.01 - 3.47i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.75T + 71T^{2} \)
73 \( 1 + 2.37T + 73T^{2} \)
79 \( 1 + (-8.80 + 5.08i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.62 + 6.28i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 5.34iT - 89T^{2} \)
97 \( 1 + (-6.24 - 10.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.002578163554806812512325912979, −8.297480724698401473690940395334, −7.75937297135682242677336922491, −6.75528532295188936819756578867, −5.84696153217729026206543557321, −5.17292165566225755456219600537, −4.12968229475188219027901664176, −3.17131780330014694596399757639, −2.02905897925084089394173579564, −0.69339152863933710561095272778, 1.51178308547729397445461668758, 2.26608696023373612616145897765, 3.60771864916944807005549439080, 4.73536988650635434784332675802, 5.26312458732632940360036231402, 6.20545554118235489697857179889, 7.21894148573328772295929386997, 7.87630068757940461021273780101, 8.752329692428958034939566585207, 9.442833069369849731397546067657

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