Properties

Label 2-2e2-4.3-c12-0-3
Degree $2$
Conductor $4$
Sign $0.793 + 0.608i$
Analytic cond. $3.65597$
Root an. cond. $1.91206$
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  + (60.6 + 20.5i)2-s − 1.28e3i·3-s + (3.24e3 + 2.49e3i)4-s + 6.16e3·5-s + (2.65e4 − 7.80e4i)6-s + 4.47e4i·7-s + (1.45e5 + 2.18e5i)8-s − 1.12e6·9-s + (3.73e5 + 1.26e5i)10-s + 1.30e6i·11-s + (3.21e6 − 4.18e6i)12-s − 7.27e5·13-s + (−9.20e5 + 2.71e6i)14-s − 7.93e6i·15-s + (4.33e6 + 1.62e7i)16-s + 1.68e7·17-s + ⋯
L(s)  = 1  + (0.946 + 0.321i)2-s − 1.76i·3-s + (0.793 + 0.608i)4-s + 0.394·5-s + (0.568 − 1.67i)6-s + 0.380i·7-s + (0.555 + 0.831i)8-s − 2.12·9-s + (0.373 + 0.126i)10-s + 0.739i·11-s + (1.07 − 1.40i)12-s − 0.150·13-s + (−0.122 + 0.359i)14-s − 0.696i·15-s + (0.258 + 0.966i)16-s + 0.697·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $0.793 + 0.608i$
Analytic conductor: \(3.65597\)
Root analytic conductor: \(1.91206\)
Motivic weight: \(12\)
Rational: no
Arithmetic: yes
Character: $\chi_{4} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4,\ (\ :6),\ 0.793 + 0.608i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(2.28111 - 0.774696i\)
\(L(\frac12)\) \(\approx\) \(2.28111 - 0.774696i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-60.6 - 20.5i)T \)
good3 \( 1 + 1.28e3iT - 5.31e5T^{2} \)
5 \( 1 - 6.16e3T + 2.44e8T^{2} \)
7 \( 1 - 4.47e4iT - 1.38e10T^{2} \)
11 \( 1 - 1.30e6iT - 3.13e12T^{2} \)
13 \( 1 + 7.27e5T + 2.32e13T^{2} \)
17 \( 1 - 1.68e7T + 5.82e14T^{2} \)
19 \( 1 - 2.86e7iT - 2.21e15T^{2} \)
23 \( 1 + 1.91e8iT - 2.19e16T^{2} \)
29 \( 1 + 6.45e8T + 3.53e17T^{2} \)
31 \( 1 + 1.21e9iT - 7.87e17T^{2} \)
37 \( 1 - 2.28e9T + 6.58e18T^{2} \)
41 \( 1 + 4.67e9T + 2.25e19T^{2} \)
43 \( 1 + 3.67e9iT - 3.99e19T^{2} \)
47 \( 1 - 1.45e10iT - 1.16e20T^{2} \)
53 \( 1 + 1.67e10T + 4.91e20T^{2} \)
59 \( 1 - 1.52e10iT - 1.77e21T^{2} \)
61 \( 1 - 2.26e10T + 2.65e21T^{2} \)
67 \( 1 - 4.19e10iT - 8.18e21T^{2} \)
71 \( 1 + 3.90e9iT - 1.64e22T^{2} \)
73 \( 1 - 1.71e11T + 2.29e22T^{2} \)
79 \( 1 + 4.28e11iT - 5.90e22T^{2} \)
83 \( 1 + 3.67e11iT - 1.06e23T^{2} \)
89 \( 1 + 3.21e10T + 2.46e23T^{2} \)
97 \( 1 - 1.07e12T + 6.93e23T^{2} \)
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

−22.50778757917168739975118801745, −20.45614729503231384498273735023, −18.66263690223322650405303496106, −17.10740258030809575714635392711, −14.56567982138910548959070610167, −13.10394757049963743784306396206, −12.00146758604148499364564888105, −7.67720139133811679677099022767, −6.03074551555212510479951314729, −2.12159529536355468119233187238, 3.54381533489840754041328877014, 5.37117840737053893438453537373, 9.800011674326689051692335821101, 11.21712170495027726050187740322, 13.89442184937612526468238388519, 15.35827818795853941977150770706, 16.69243893370555579427613934248, 19.87384580797708946356666328012, 21.25080779943412573494076763276, 21.91854279806780635997479442456

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