Properties

Label 2-3-3.2-c12-0-2
Degree $2$
Conductor $3$
Sign $-0.925 + 0.377i$
Analytic cond. $2.74198$
Root an. cond. $1.65589$
Motivic weight $12$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 91.7i·2-s + (−675 + 275. i)3-s − 4.32e3·4-s − 2.11e4i·5-s + (2.52e4 + 6.19e4i)6-s + 4.02e4·7-s + 2.12e4i·8-s + (3.79e5 − 3.71e5i)9-s − 1.93e6·10-s + 1.16e6i·11-s + (2.92e6 − 1.19e6i)12-s + 1.28e6·13-s − 3.69e6i·14-s + (5.81e6 + 1.42e7i)15-s − 1.57e7·16-s − 1.48e7i·17-s + ⋯
L(s)  = 1  − 1.43i·2-s + (−0.925 + 0.377i)3-s − 1.05·4-s − 1.35i·5-s + (0.541 + 1.32i)6-s + 0.342·7-s + 0.0812i·8-s + (0.714 − 0.699i)9-s − 1.93·10-s + 0.655i·11-s + (0.978 − 0.399i)12-s + 0.266·13-s − 0.490i·14-s + (0.510 + 1.25i)15-s − 0.940·16-s − 0.614i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.925 + 0.377i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.925 + 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $-0.925 + 0.377i$
Analytic conductor: \(2.74198\)
Root analytic conductor: \(1.65589\)
Motivic weight: \(12\)
Rational: no
Arithmetic: yes
Character: $\chi_{3} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3,\ (\ :6),\ -0.925 + 0.377i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(0.188276 - 0.960023i\)
\(L(\frac12)\) \(\approx\) \(0.188276 - 0.960023i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (675 - 275. i)T \)
good2 \( 1 + 91.7iT - 4.09e3T^{2} \)
5 \( 1 + 2.11e4iT - 2.44e8T^{2} \)
7 \( 1 - 4.02e4T + 1.38e10T^{2} \)
11 \( 1 - 1.16e6iT - 3.13e12T^{2} \)
13 \( 1 - 1.28e6T + 2.32e13T^{2} \)
17 \( 1 + 1.48e7iT - 5.82e14T^{2} \)
19 \( 1 - 5.33e7T + 2.21e15T^{2} \)
23 \( 1 + 1.07e8iT - 2.19e16T^{2} \)
29 \( 1 + 1.20e8iT - 3.53e17T^{2} \)
31 \( 1 - 6.65e7T + 7.87e17T^{2} \)
37 \( 1 - 2.22e9T + 6.58e18T^{2} \)
41 \( 1 - 8.21e9iT - 2.25e19T^{2} \)
43 \( 1 - 8.97e9T + 3.99e19T^{2} \)
47 \( 1 - 1.07e9iT - 1.16e20T^{2} \)
53 \( 1 + 4.11e10iT - 4.91e20T^{2} \)
59 \( 1 - 4.61e10iT - 1.77e21T^{2} \)
61 \( 1 + 4.06e10T + 2.65e21T^{2} \)
67 \( 1 - 1.21e11T + 8.18e21T^{2} \)
71 \( 1 + 4.48e10iT - 1.64e22T^{2} \)
73 \( 1 + 6.09e10T + 2.29e22T^{2} \)
79 \( 1 + 2.52e11T + 5.90e22T^{2} \)
83 \( 1 - 4.10e11iT - 1.06e23T^{2} \)
89 \( 1 + 1.12e11iT - 2.46e23T^{2} \)
97 \( 1 - 6.53e11T + 6.93e23T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.69971702967702057995006089881, −21.12006835019698744233446763829, −20.24456941976864783445144189082, −18.03274477295533426490251460125, −16.23295800461152333481842013949, −12.77697009445635526237317130872, −11.54389191159432437564054235340, −9.618781225369489204210340327025, −4.66217614320510640493547412182, −0.981697946157363364975853116385, 5.91740399460202559386893852271, 7.42240927695159248789973113821, 11.10897317795989106351748506705, 14.06103303664286913266335788310, 15.75789095111368458466305609757, 17.43035593622051377478465341535, 18.61987516488622206935257128072, 22.07097712265710489133444388825, 23.28240514060699706949461507467, 24.46060255202734799575596320719

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