↑

Properties

Label 2-2e2-4.3-c10-0-2
Degree $2$
Conductor $4$
Sign $-0.267 + 0.963i$
Analytic cond. $2.54142$
Root an. cond. $1.59418$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−25.4 + 19.3i)2-s − 343. i·3-s + (273. − 986. i)4-s − 3.26e3·5-s + (6.65e3 + 8.75e3i)6-s − 1.09e4i·7-s + (1.21e4 + 3.04e4i)8-s − 5.89e4·9-s + (8.32e4 − 6.32e4i)10-s − 6.44e4i·11-s + (−3.38e5 − 9.40e4i)12-s + 4.06e5·13-s + (2.11e5 + 2.77e5i)14-s + 1.12e6i·15-s + (−8.98e5 − 5.40e5i)16-s + 6.64e4·17-s + ⋯
L(s)  = 1  + (−0.796 + 0.605i)2-s − 1.41i·3-s + (0.267 − 0.963i)4-s − 1.04·5-s + (0.855 + 1.12i)6-s − 0.648i·7-s + (0.370 + 0.928i)8-s − 0.998·9-s + (0.832 − 0.632i)10-s − 0.400i·11-s + (−1.36 − 0.377i)12-s + 1.09·13-s + (0.392 + 0.516i)14-s + 1.47i·15-s + (−0.857 − 0.515i)16-s + 0.0468·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.267 + 0.963i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.267 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $-0.267 + 0.963i$
Analytic conductor: \(2.54142\)
Root analytic conductor: \(1.59418\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{4} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4,\ (\ :5),\ -0.267 + 0.963i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.414301 - 0.544842i\)
\(L(\frac12)\) \(\approx\) \(0.414301 - 0.544842i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (25.4 - 19.3i)T \)
good3 \( 1 + 343. iT - 5.90e4T^{2} \)
5 \( 1 + 3.26e3T + 9.76e6T^{2} \)
7 \( 1 + 1.09e4iT - 2.82e8T^{2} \)
11 \( 1 + 6.44e4iT - 2.59e10T^{2} \)
13 \( 1 - 4.06e5T + 1.37e11T^{2} \)
17 \( 1 - 6.64e4T + 2.01e12T^{2} \)
19 \( 1 + 3.81e6iT - 6.13e12T^{2} \)
23 \( 1 - 3.74e6iT - 4.14e13T^{2} \)
29 \( 1 - 2.15e7T + 4.20e14T^{2} \)
31 \( 1 - 1.91e6iT - 8.19e14T^{2} \)
37 \( 1 - 1.00e7T + 4.80e15T^{2} \)
41 \( 1 + 1.45e8T + 1.34e16T^{2} \)
43 \( 1 - 6.57e7iT - 2.16e16T^{2} \)
47 \( 1 - 1.77e8iT - 5.25e16T^{2} \)
53 \( 1 - 1.53e8T + 1.74e17T^{2} \)
59 \( 1 + 1.41e9iT - 5.11e17T^{2} \)
61 \( 1 - 1.31e9T + 7.13e17T^{2} \)
67 \( 1 - 2.19e9iT - 1.82e18T^{2} \)
71 \( 1 + 2.04e9iT - 3.25e18T^{2} \)
73 \( 1 - 1.76e9T + 4.29e18T^{2} \)
79 \( 1 - 1.22e9iT - 9.46e18T^{2} \)
83 \( 1 - 2.60e9iT - 1.55e19T^{2} \)
89 \( 1 + 1.25e8T + 3.11e19T^{2} \)
97 \( 1 - 5.70e9T + 7.37e19T^{2} \)
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

−23.43567158180110024651479294609, −19.91943465218265821640063986107, −18.97162843670759311254509711095, −17.63150725769856556687939751625, −15.79700897927098509864935303123, −13.61499643036649530810518495762, −11.33789846092909048957911544556, −8.208786210900433162932622111621, −6.83984335762873511449215058431, −0.78089318857218356909645083674, 3.81673543849896934381827652781, 8.519834491300905555941675999667, 10.32368575272786140735334095153, 11.91760884222305783945399268157, 15.39420473943673807893529768838, 16.42017974295603327465488566857, 18.60429715559805218345681938475, 20.26819393586697034059751448452, 21.36112204049376122437775465006, 22.84209775503130548786495359525

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