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Properties

Label 2-12e3-8.5-c3-0-31
Degree $2$
Conductor $1728$
Sign $-0.707 + 0.707i$
Analytic cond. $101.955$
Root an. cond. $10.0972$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 16.7i·5-s − 22.3·7-s + 51.9i·11-s + 28.0i·13-s − 60.9·17-s + 162. i·19-s + 83.2·23-s − 156.·25-s + 217. i·29-s − 174.·31-s − 374. i·35-s + 407. i·37-s + 196.·41-s − 345. i·43-s + 335.·47-s + ⋯
L(s)  = 1  + 1.50i·5-s − 1.20·7-s + 1.42i·11-s + 0.598i·13-s − 0.869·17-s + 1.96i·19-s + 0.755·23-s − 1.25·25-s + 1.39i·29-s − 1.00·31-s − 1.80i·35-s + 1.81i·37-s + 0.748·41-s − 1.22i·43-s + 1.04·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-0.707 + 0.707i$
Analytic conductor: \(101.955\)
Root analytic conductor: \(10.0972\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :3/2),\ -0.707 + 0.707i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.303411111\)
\(L(\frac12)\) \(\approx\) \(1.303411111\)
\(L(\frac{5}{2})\) 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 \( 1 - 16.7iT - 125T^{2} \)
7 \( 1 + 22.3T + 343T^{2} \)
11 \( 1 - 51.9iT - 1.33e3T^{2} \)
13 \( 1 - 28.0iT - 2.19e3T^{2} \)
17 \( 1 + 60.9T + 4.91e3T^{2} \)
19 \( 1 - 162. iT - 6.85e3T^{2} \)
23 \( 1 - 83.2T + 1.21e4T^{2} \)
29 \( 1 - 217. iT - 2.43e4T^{2} \)
31 \( 1 + 174.T + 2.97e4T^{2} \)
37 \( 1 - 407. iT - 5.06e4T^{2} \)
41 \( 1 - 196.T + 6.89e4T^{2} \)
43 \( 1 + 345. iT - 7.95e4T^{2} \)
47 \( 1 - 335.T + 1.03e5T^{2} \)
53 \( 1 - 234. iT - 1.48e5T^{2} \)
59 \( 1 - 321. iT - 2.05e5T^{2} \)
61 \( 1 - 739. iT - 2.26e5T^{2} \)
67 \( 1 - 20iT - 3.00e5T^{2} \)
71 \( 1 - 418.T + 3.57e5T^{2} \)
73 \( 1 - 397.T + 3.89e5T^{2} \)
79 \( 1 + 95.6T + 4.93e5T^{2} \)
83 \( 1 + 348. iT - 5.71e5T^{2} \)
89 \( 1 + 1.49e3T + 7.04e5T^{2} \)
97 \( 1 - 881.T + 9.12e5T^{2} \)
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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

−9.577128628719686480991054262460, −8.836920818292418694175125628916, −7.52075722946254709233532355962, −6.97135677668562255129397492829, −6.52734199354150221451143384905, −5.60282160444387977790038351910, −4.28353738210744137777823967866, −3.49345703755709058970069762361, −2.65517626130723907736959311923, −1.68189001009827484078709314313, 0.47624660734728009801307157406, 0.58591660366425965231291782859, 2.37802664525908101411225911726, 3.35574997634629955482822032451, 4.33187526118565064534248715183, 5.23860623921988833025804796941, 5.94706755214877568616704661951, 6.76857812301981003085333217889, 7.81265233451948117035856941403, 8.726473280986648999716693562893

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