↑

Properties

Label 2-2e2-4.3-c16-0-0
Degree $2$
Conductor $4$
Sign $-0.619 - 0.785i$
Analytic cond. $6.49298$
Root an. cond. $2.54813$
Motivic weight $16$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (111. − 230. i)2-s + 8.02e3i·3-s + (−4.05e4 − 5.14e4i)4-s − 6.64e5·5-s + (1.84e6 + 8.96e5i)6-s + 6.08e6i·7-s + (−1.63e7 + 3.59e6i)8-s − 2.13e7·9-s + (−7.41e7 + 1.52e8i)10-s − 8.45e7i·11-s + (4.12e8 − 3.25e8i)12-s − 2.60e8·13-s + (1.40e9 + 6.80e8i)14-s − 5.32e9i·15-s + (−1.00e9 + 4.17e9i)16-s − 2.57e9·17-s + ⋯
L(s)  = 1  + (0.436 − 0.899i)2-s + 1.22i·3-s + (−0.619 − 0.785i)4-s − 1.70·5-s + (1.10 + 0.533i)6-s + 1.05i·7-s + (−0.976 + 0.214i)8-s − 0.495·9-s + (−0.741 + 1.52i)10-s − 0.394i·11-s + (0.960 − 0.757i)12-s − 0.319·13-s + (0.950 + 0.460i)14-s − 2.07i·15-s + (−0.233 + 0.972i)16-s − 0.368·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.619 - 0.785i)\, \overline{\Lambda}(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (-0.619 - 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $-0.619 - 0.785i$
Analytic conductor: \(6.49298\)
Root analytic conductor: \(2.54813\)
Motivic weight: \(16\)
Rational: no
Arithmetic: yes
Character: $\chi_{4} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4,\ (\ :8),\ -0.619 - 0.785i)\)

Particular Values

\(L(\frac{17}{2})\) \(\approx\) \(0.229085 + 0.472313i\)
\(L(\frac12)\) \(\approx\) \(0.229085 + 0.472313i\)
\(L(9)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-111. + 230. i)T \)
good3 \( 1 - 8.02e3iT - 4.30e7T^{2} \)
5 \( 1 + 6.64e5T + 1.52e11T^{2} \)
7 \( 1 - 6.08e6iT - 3.32e13T^{2} \)
11 \( 1 + 8.45e7iT - 4.59e16T^{2} \)
13 \( 1 + 2.60e8T + 6.65e17T^{2} \)
17 \( 1 + 2.57e9T + 4.86e19T^{2} \)
19 \( 1 + 5.86e9iT - 2.88e20T^{2} \)
23 \( 1 + 1.86e10iT - 6.13e21T^{2} \)
29 \( 1 + 4.45e11T + 2.50e23T^{2} \)
31 \( 1 - 1.48e12iT - 7.27e23T^{2} \)
37 \( 1 + 1.01e11T + 1.23e25T^{2} \)
41 \( 1 + 8.62e12T + 6.37e25T^{2} \)
43 \( 1 + 4.49e12iT - 1.36e26T^{2} \)
47 \( 1 - 3.73e13iT - 5.66e26T^{2} \)
53 \( 1 - 2.92e13T + 3.87e27T^{2} \)
59 \( 1 + 1.24e14iT - 2.15e28T^{2} \)
61 \( 1 + 3.30e14T + 3.67e28T^{2} \)
67 \( 1 - 1.53e14iT - 1.64e29T^{2} \)
71 \( 1 + 8.32e14iT - 4.16e29T^{2} \)
73 \( 1 - 2.42e14T + 6.50e29T^{2} \)
79 \( 1 - 1.13e15iT - 2.30e30T^{2} \)
83 \( 1 + 2.68e15iT - 5.07e30T^{2} \)
89 \( 1 - 4.30e15T + 1.54e31T^{2} \)
97 \( 1 - 2.79e15T + 6.14e31T^{2} \)
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

−21.55940571615594714793352278919, −20.07914313857061514683343727779, −18.81929408382061790208829234847, −15.84312738639965431070438223724, −14.95591658404535612702219454749, −12.19372132229483081381658412857, −10.91660335070179241609302721300, −8.909540015188048667350505884078, −4.79308891582340723749629852904, −3.34387080671108873452417632637, 0.25717048327944685822988837240, 4.05345917183255397365910737805, 7.07047661210979111301884939177, 7.86485585409566548849748363107, 11.91784814888708944183800398649, 13.31116822130481623459288317046, 15.14743178609489468992357203573, 16.83577262275187315599610108336, 18.59575218958299145945891565713, 20.04880483998241323481620558771

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