↑

Properties

Label 2-14e2-28.27-c5-0-22
Degree $2$
Conductor $196$
Sign $-0.396 - 0.917i$
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  + (1.57 + 5.43i)2-s − 7.64·3-s + (−27.0 + 17.1i)4-s − 63.9i·5-s + (−12.0 − 41.5i)6-s + (−135. − 119. i)8-s − 184.·9-s + (347. − 100. i)10-s − 601. i·11-s + (206. − 130. i)12-s + 890. i·13-s + 488. i·15-s + (436. − 926. i)16-s + 116. i·17-s + (−291. − 1.00e3i)18-s + 901.·19-s + ⋯
L(s)  = 1  + (0.278 + 0.960i)2-s − 0.490·3-s + (−0.844 + 0.535i)4-s − 1.14i·5-s + (−0.136 − 0.470i)6-s + (−0.749 − 0.661i)8-s − 0.759·9-s + (1.09 − 0.318i)10-s − 1.49i·11-s + (0.414 − 0.262i)12-s + 1.46i·13-s + 0.560i·15-s + (0.426 − 0.904i)16-s + 0.0978i·17-s + (−0.211 − 0.729i)18-s + 0.573·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.396 - 0.917i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.396 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $-0.396 - 0.917i$
Analytic conductor: \(31.4352\)
Root analytic conductor: \(5.60671\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{196} (195, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 196,\ (\ :5/2),\ -0.396 - 0.917i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.085995177\)
\(L(\frac12)\) \(\approx\) \(1.085995177\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.57 - 5.43i)T \)
7 \( 1 \)
good3 \( 1 + 7.64T + 243T^{2} \)
5 \( 1 + 63.9iT - 3.12e3T^{2} \)
11 \( 1 + 601. iT - 1.61e5T^{2} \)
13 \( 1 - 890. iT - 3.71e5T^{2} \)
17 \( 1 - 116. iT - 1.41e6T^{2} \)
19 \( 1 - 901.T + 2.47e6T^{2} \)
23 \( 1 - 3.88e3iT - 6.43e6T^{2} \)
29 \( 1 - 124.T + 2.05e7T^{2} \)
31 \( 1 + 7.11T + 2.86e7T^{2} \)
37 \( 1 - 1.50e3T + 6.93e7T^{2} \)
41 \( 1 - 1.93e4iT - 1.15e8T^{2} \)
43 \( 1 + 2.30e3iT - 1.47e8T^{2} \)
47 \( 1 + 8.62e3T + 2.29e8T^{2} \)
53 \( 1 + 2.81e4T + 4.18e8T^{2} \)
59 \( 1 - 5.26e4T + 7.14e8T^{2} \)
61 \( 1 - 2.39e4iT - 8.44e8T^{2} \)
67 \( 1 + 4.95e4iT - 1.35e9T^{2} \)
71 \( 1 - 6.72e4iT - 1.80e9T^{2} \)
73 \( 1 + 3.96e4iT - 2.07e9T^{2} \)
79 \( 1 + 5.60e4iT - 3.07e9T^{2} \)
83 \( 1 - 1.42e4T + 3.93e9T^{2} \)
89 \( 1 - 1.19e5iT - 5.58e9T^{2} \)
97 \( 1 - 5.12e4iT - 8.58e9T^{2} \)
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

−11.87233925823795715813503386961, −11.36252788807899249368540324517, −9.479556658490599227401086332961, −8.791404845899280847693358270828, −7.954128268491671785299760844120, −6.49102384024320324715806839407, −5.62619729346174372133916884929, −4.78748806528215245130252010779, −3.43739564416935754928894537421, −0.958844049510372326062902434849, 0.41987500096200220108342475654, 2.32588440946119157417094683994, 3.24191351276746786130093548786, 4.77389698495591046463947933804, 5.82505388786763187386612448502, 7.04189954314529672885548203758, 8.403049599989077376035090595444, 9.846200398387965027757524390729, 10.49026189725845663808042123998, 11.21065235180953759858625225690

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