↑

Properties

Label 2-392-56.13-c2-0-20
Degree $2$
Conductor $392$
Sign $-0.861 - 0.508i$
Analytic cond. $10.6812$
Root an. cond. $3.26821$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.82 + 0.807i)2-s + 0.910·3-s + (2.69 + 2.95i)4-s − 6.34·5-s + (1.66 + 0.735i)6-s + (2.54 + 7.58i)8-s − 8.17·9-s + (−11.6 − 5.12i)10-s + 13.2i·11-s + (2.45 + 2.69i)12-s − 19.4·13-s − 5.77·15-s + (−1.46 + 15.9i)16-s + 15.9i·17-s + (−14.9 − 6.59i)18-s + 16.4·19-s + ⋯
L(s)  = 1  + (0.914 + 0.403i)2-s + 0.303·3-s + (0.673 + 0.738i)4-s − 1.26·5-s + (0.277 + 0.122i)6-s + (0.318 + 0.948i)8-s − 0.907·9-s + (−1.16 − 0.512i)10-s + 1.20i·11-s + (0.204 + 0.224i)12-s − 1.49·13-s − 0.385·15-s + (−0.0917 + 0.995i)16-s + 0.936i·17-s + (−0.830 − 0.366i)18-s + 0.866·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.861 - 0.508i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.861 - 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $-0.861 - 0.508i$
Analytic conductor: \(10.6812\)
Root analytic conductor: \(3.26821\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1),\ -0.861 - 0.508i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.432070 + 1.58178i\)
\(L(\frac12)\) \(\approx\) \(0.432070 + 1.58178i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.82 - 0.807i)T \)
7 \( 1 \)
good3 \( 1 - 0.910T + 9T^{2} \)
5 \( 1 + 6.34T + 25T^{2} \)
11 \( 1 - 13.2iT - 121T^{2} \)
13 \( 1 + 19.4T + 169T^{2} \)
17 \( 1 - 15.9iT - 289T^{2} \)
19 \( 1 - 16.4T + 361T^{2} \)
23 \( 1 - 23.9T + 529T^{2} \)
29 \( 1 - 16.6iT - 841T^{2} \)
31 \( 1 + 12.8iT - 961T^{2} \)
37 \( 1 + 47.5iT - 1.36e3T^{2} \)
41 \( 1 + 6.49iT - 1.68e3T^{2} \)
43 \( 1 - 33.2iT - 1.84e3T^{2} \)
47 \( 1 - 21.9iT - 2.20e3T^{2} \)
53 \( 1 + 37.1iT - 2.80e3T^{2} \)
59 \( 1 - 54.6T + 3.48e3T^{2} \)
61 \( 1 - 10.2T + 3.72e3T^{2} \)
67 \( 1 + 17.1iT - 4.48e3T^{2} \)
71 \( 1 - 32.0T + 5.04e3T^{2} \)
73 \( 1 - 107. iT - 5.32e3T^{2} \)
79 \( 1 + 58.3T + 6.24e3T^{2} \)
83 \( 1 + 36.3T + 6.88e3T^{2} \)
89 \( 1 + 1.07iT - 7.92e3T^{2} \)
97 \( 1 - 169. iT - 9.40e3T^{2} \)
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

−11.72888455367589324448303398424, −10.92374171310032631210238502192, −9.552790359701181597368334662273, −8.346308863672760098758086014953, −7.54708734365321032935919805640, −6.98665869157694242492018456407, −5.46918884059451295454222907497, −4.56453353053843930011689631859, −3.54767000476806952975420935033, −2.42365882940944918741652905722, 0.48580945060836729728442024824, 2.77278571456556963940071039278, 3.38025539603187974011644752758, 4.71107254556794558595456765325, 5.56221677747442813531193428678, 6.96597459836152783884872506572, 7.76466963962963632829536124081, 8.858233137742280656484742342130, 9.965723136422669492462805848527, 11.20771710887230644431381515910

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