↑

Properties

Label 2-14e2-7.5-c8-0-23
Degree $2$
Conductor $196$
Sign $-0.895 - 0.444i$
Analytic cond. $79.8462$
Root an. cond. $8.93567$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−67.7 + 39.0i)3-s + (−594. − 342. i)5-s + (−222. + 386. i)9-s + (−1.22e4 − 2.12e4i)11-s + 275. i·13-s + 5.36e4·15-s + (1.33e5 − 7.72e4i)17-s + (−9.69e3 − 5.59e3i)19-s + (−1.42e5 + 2.47e5i)23-s + (3.99e4 + 6.91e4i)25-s − 5.47e5i·27-s − 2.66e5·29-s + (1.28e6 − 7.39e5i)31-s + (1.66e6 + 9.58e5i)33-s + (1.24e6 − 2.15e6i)37-s + ⋯
L(s)  = 1  + (−0.836 + 0.482i)3-s + (−0.950 − 0.548i)5-s + (−0.0339 + 0.0588i)9-s + (−0.837 − 1.45i)11-s + 0.00964i·13-s + 1.05·15-s + (1.60 − 0.924i)17-s + (−0.0743 − 0.0429i)19-s + (−0.510 + 0.883i)23-s + (0.102 + 0.177i)25-s − 1.03i·27-s − 0.376·29-s + (1.38 − 0.800i)31-s + (1.40 + 0.808i)33-s + (0.664 − 1.15i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 - 0.444i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $-0.895 - 0.444i$
Analytic conductor: \(79.8462\)
Root analytic conductor: \(8.93567\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{196} (117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 196,\ (\ :4),\ -0.895 - 0.444i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.1502844906\)
\(L(\frac12)\) \(\approx\) \(0.1502844906\)
\(L(5)\) 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 \( 1 + (67.7 - 39.0i)T + (3.28e3 - 5.68e3i)T^{2} \)
5 \( 1 + (594. + 342. i)T + (1.95e5 + 3.38e5i)T^{2} \)
11 \( 1 + (1.22e4 + 2.12e4i)T + (-1.07e8 + 1.85e8i)T^{2} \)
13 \( 1 - 275. iT - 8.15e8T^{2} \)
17 \( 1 + (-1.33e5 + 7.72e4i)T + (3.48e9 - 6.04e9i)T^{2} \)
19 \( 1 + (9.69e3 + 5.59e3i)T + (8.49e9 + 1.47e10i)T^{2} \)
23 \( 1 + (1.42e5 - 2.47e5i)T + (-3.91e10 - 6.78e10i)T^{2} \)
29 \( 1 + 2.66e5T + 5.00e11T^{2} \)
31 \( 1 + (-1.28e6 + 7.39e5i)T + (4.26e11 - 7.38e11i)T^{2} \)
37 \( 1 + (-1.24e6 + 2.15e6i)T + (-1.75e12 - 3.04e12i)T^{2} \)
41 \( 1 - 7.08e5iT - 7.98e12T^{2} \)
43 \( 1 + 1.28e6T + 1.16e13T^{2} \)
47 \( 1 + (-1.44e6 - 8.36e5i)T + (1.19e13 + 2.06e13i)T^{2} \)
53 \( 1 + (-9.80e5 - 1.69e6i)T + (-3.11e13 + 5.39e13i)T^{2} \)
59 \( 1 + (-1.09e7 + 6.34e6i)T + (7.34e13 - 1.27e14i)T^{2} \)
61 \( 1 + (4.25e6 + 2.45e6i)T + (9.58e13 + 1.66e14i)T^{2} \)
67 \( 1 + (1.33e7 + 2.31e7i)T + (-2.03e14 + 3.51e14i)T^{2} \)
71 \( 1 + 5.66e6T + 6.45e14T^{2} \)
73 \( 1 + (3.59e7 - 2.07e7i)T + (4.03e14 - 6.98e14i)T^{2} \)
79 \( 1 + (1.42e7 - 2.47e7i)T + (-7.58e14 - 1.31e15i)T^{2} \)
83 \( 1 + 6.35e7iT - 2.25e15T^{2} \)
89 \( 1 + (-1.00e7 - 5.81e6i)T + (1.96e15 + 3.40e15i)T^{2} \)
97 \( 1 - 8.14e6iT - 7.83e15T^{2} \)
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

−10.64037296352853143206631692560, −9.616640518716857522506934407608, −8.217613448909619905273968489589, −7.69344032057651703170626003570, −5.89654204714893225766208711169, −5.27127002151531525052910136151, −4.11717425166018958886831806937, −2.91886435862564591613016212813, −0.795155826089073851523715730200, −0.05872720632577880992671975211, 1.31540119504857827749257056085, 2.89630281337034955645266028553, 4.20112440059873330968607953977, 5.43948259933878222148963741016, 6.57536422115702429680674726420, 7.45421141203177962376052316038, 8.255843116836379232038602237040, 9.979160993434106757003181082906, 10.61764782128530188561074701546, 11.91706090965031852235471884388

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