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Properties

Label 2-12e3-36.7-c2-0-25
Degree $2$
Conductor $1728$
Sign $0.982 - 0.186i$
Analytic cond. $47.0845$
Root an. cond. $6.86182$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.35 + 2.34i)5-s + (10.0 + 5.79i)7-s + (−8.54 − 4.93i)11-s + (−0.296 − 0.513i)13-s + 8.87·17-s − 14.0i·19-s + (18.2 − 10.5i)23-s + (8.82 − 15.2i)25-s + (10.1 − 17.6i)29-s + (14.3 − 8.27i)31-s + 31.4i·35-s + 40.6·37-s + (−21.2 − 36.7i)41-s + (32.2 + 18.6i)43-s + (−1.57 − 0.907i)47-s + ⋯
L(s)  = 1  + (0.271 + 0.469i)5-s + (1.43 + 0.828i)7-s + (−0.777 − 0.448i)11-s + (−0.0227 − 0.0394i)13-s + 0.522·17-s − 0.742i·19-s + (0.794 − 0.458i)23-s + (0.352 − 0.611i)25-s + (0.350 − 0.607i)29-s + (0.462 − 0.266i)31-s + 0.898i·35-s + 1.09·37-s + (−0.517 − 0.896i)41-s + (0.749 + 0.432i)43-s + (−0.0334 − 0.0193i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.186i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.982 - 0.186i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.982 - 0.186i$
Analytic conductor: \(47.0845\)
Root analytic conductor: \(6.86182\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1),\ 0.982 - 0.186i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.631529187\)
\(L(\frac12)\) \(\approx\) \(2.631529187\)
\(L(2)\) 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 \( 1 + (-1.35 - 2.34i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (-10.0 - 5.79i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (8.54 + 4.93i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (0.296 + 0.513i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 - 8.87T + 289T^{2} \)
19 \( 1 + 14.0iT - 361T^{2} \)
23 \( 1 + (-18.2 + 10.5i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-10.1 + 17.6i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (-14.3 + 8.27i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 - 40.6T + 1.36e3T^{2} \)
41 \( 1 + (21.2 + 36.7i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-32.2 - 18.6i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (1.57 + 0.907i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 21.1T + 2.80e3T^{2} \)
59 \( 1 + (-76.6 + 44.2i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (36.4 - 63.2i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-38.3 + 22.1i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 111. iT - 5.04e3T^{2} \)
73 \( 1 + 76.2T + 5.32e3T^{2} \)
79 \( 1 + (8.30 + 4.79i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-73.6 - 42.5i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 64.7T + 7.92e3T^{2} \)
97 \( 1 + (3.59 - 6.22i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−9.011389513810267994103581993980, −8.319193106212845557599203166157, −7.76360681417203824303537150592, −6.75463250706584008337662850208, −5.79468124404246987683695779937, −5.14157595650859808454279149722, −4.34535740741312771331920945735, −2.83017989587920964661908938777, −2.31899506577021257293228242741, −0.898317860190383048832897532570, 1.00229021932008264780434647622, 1.76911424712179096315048078091, 3.11868716455389593304523496984, 4.34506955598628946725427806847, 4.97243286081163299428051818956, 5.62523953355446421280692390051, 6.90387502919823058756082839411, 7.69644407616341194290406935043, 8.142420731740603171097588745385, 9.074263346815617258869692093662

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