↑

Properties

Label 2-14e2-7.4-c5-0-4
Degree $2$
Conductor $196$
Sign $-0.968 - 0.250i$
Analytic cond. $31.4352$
Root an. cond. $5.60671$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−6 + 10.3i)3-s + (27 + 46.7i)5-s + (49.5 + 85.7i)9-s + (−270 + 467. i)11-s + 418·13-s − 648·15-s + (297 − 514. i)17-s + (418 + 723. i)19-s + (2.05e3 + 3.55e3i)23-s + (104.5 − 180. i)25-s − 4.10e3·27-s − 594·29-s + (2.12e3 − 3.68e3i)31-s + (−3.24e3 − 5.61e3i)33-s + (149 + 258. i)37-s + ⋯
L(s)  = 1  + (−0.384 + 0.666i)3-s + (0.482 + 0.836i)5-s + (0.203 + 0.352i)9-s + (−0.672 + 1.16i)11-s + 0.685·13-s − 0.743·15-s + (0.249 − 0.431i)17-s + (0.265 + 0.460i)19-s + (0.808 + 1.40i)23-s + (0.0334 − 0.0579i)25-s − 1.08·27-s − 0.131·29-s + (0.397 − 0.688i)31-s + (−0.517 − 0.897i)33-s + (0.0178 + 0.0309i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $-0.968 - 0.250i$
Analytic conductor: \(31.4352\)
Root analytic conductor: \(5.60671\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{196} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 196,\ (\ :5/2),\ -0.968 - 0.250i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.479199893\)
\(L(\frac12)\) \(\approx\) \(1.479199893\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (6 - 10.3i)T + (-121.5 - 210. i)T^{2} \)
5 \( 1 + (-27 - 46.7i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (270 - 467. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 - 418T + 3.71e5T^{2} \)
17 \( 1 + (-297 + 514. i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-418 - 723. i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (-2.05e3 - 3.55e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + 594T + 2.05e7T^{2} \)
31 \( 1 + (-2.12e3 + 3.68e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (-149 - 258. i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + 1.72e4T + 1.15e8T^{2} \)
43 \( 1 + 1.21e4T + 1.47e8T^{2} \)
47 \( 1 + (648 + 1.12e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (9.74e3 - 1.68e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (3.83e3 - 6.64e3i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (1.73e4 + 3.00e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (1.09e4 - 1.88e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 4.68e4T + 1.80e9T^{2} \)
73 \( 1 + (-3.37e4 + 5.85e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-3.84e4 - 6.66e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 6.77e4T + 3.93e9T^{2} \)
89 \( 1 + (-1.48e4 - 2.57e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 1.22e5T + 8.58e9T^{2} \)
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

−11.85119107672026696646535185627, −10.87882798990519051768630790944, −10.14860774335752357848964180539, −9.503386557880415555607349487656, −7.87666714272711221776639454973, −6.92526924955320658115343795397, −5.64245290954351691803255546160, −4.68977070612841135500151565035, −3.24064691863337616706659543766, −1.79740224768445110230891669265, 0.49213098530591072928089558796, 1.45734951064192139503349566474, 3.24102611726558864429364232170, 4.92465214809842891543103173085, 5.93142795273266138445825588925, 6.84012529087407548993483271901, 8.279499185353390168756216445291, 8.950271390804328155913123521383, 10.26348006073767225104631589088, 11.26209083439098568390253517226

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