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Properties

Label 2-12e3-36.7-c2-0-35
Degree $2$
Conductor $1728$
Sign $-0.224 + 0.974i$
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  + (3.07 + 5.32i)5-s + (−0.511 − 0.295i)7-s + (−15.1 − 8.72i)11-s + (0.892 + 1.54i)13-s + 16.9·17-s − 19.5i·19-s + (−6.86 + 3.96i)23-s + (−6.39 + 11.0i)25-s + (3.17 − 5.49i)29-s + (−27.6 + 15.9i)31-s − 3.63i·35-s − 58.2·37-s + (2.66 + 4.62i)41-s + (−33.9 − 19.5i)43-s + (−9.64 − 5.56i)47-s + ⋯
L(s)  = 1  + (0.614 + 1.06i)5-s + (−0.0730 − 0.0421i)7-s + (−1.37 − 0.793i)11-s + (0.0686 + 0.118i)13-s + 0.995·17-s − 1.02i·19-s + (−0.298 + 0.172i)23-s + (−0.255 + 0.443i)25-s + (0.109 − 0.189i)29-s + (−0.892 + 0.515i)31-s − 0.103i·35-s − 1.57·37-s + (0.0651 + 0.112i)41-s + (−0.789 − 0.455i)43-s + (−0.205 − 0.118i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.224 + 0.974i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.224 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-0.224 + 0.974i$
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: \((2,\ 1728,\ (\ :1),\ -0.224 + 0.974i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9252640387\)
\(L(\frac12)\) \(\approx\) \(0.9252640387\)
\(L(2)\) 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 \( 1 + (-3.07 - 5.32i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (0.511 + 0.295i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (15.1 + 8.72i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-0.892 - 1.54i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 - 16.9T + 289T^{2} \)
19 \( 1 + 19.5iT - 361T^{2} \)
23 \( 1 + (6.86 - 3.96i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-3.17 + 5.49i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (27.6 - 15.9i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + 58.2T + 1.36e3T^{2} \)
41 \( 1 + (-2.66 - 4.62i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (33.9 + 19.5i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (9.64 + 5.56i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 35.8T + 2.80e3T^{2} \)
59 \( 1 + (-20.8 + 12.0i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-37.9 + 65.7i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (31.8 - 18.3i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 87.8iT - 5.04e3T^{2} \)
73 \( 1 + 60.0T + 5.32e3T^{2} \)
79 \( 1 + (32.1 + 18.5i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-66.0 - 38.1i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 27.5T + 7.92e3T^{2} \)
97 \( 1 + (-13.0 + 22.6i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−8.887192696470227271059645834296, −8.094789408264300192491493146336, −7.22571411864075571937128743845, −6.57102608138491737855164563210, −5.61373509053186602411284930866, −5.06856262439166071157761890369, −3.51661308453949902490845098821, −2.92027276174051732646450267827, −1.92125174292803829960118368902, −0.23551794440637781388705759848, 1.27794087154710350581077675613, 2.21254363821660602718311964442, 3.43683201982095703517915342930, 4.59514930389382649961684696788, 5.39353279585735186851899768465, 5.80022362720682297698273549794, 7.11243136506889458069692703433, 7.88547979944853394853705096174, 8.533772600712686160338014013366, 9.428037787332650768319046972312

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