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Properties

Label 2-14e2-7.3-c8-0-16
Degree $2$
Conductor $196$
Sign $0.611 - 0.791i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (95.3 + 55.0i)3-s + (20.9 − 12.0i)5-s + (2.77e3 + 4.81e3i)9-s + (5.76e3 − 9.98e3i)11-s + 1.61e4i·13-s + 2.65e3·15-s + (1.18e4 + 6.82e3i)17-s + (9.56e4 − 5.52e4i)19-s + (4.77e4 + 8.27e4i)23-s + (−1.95e5 + 3.37e5i)25-s − 1.10e5i·27-s + 9.68e5·29-s + (−5.56e5 − 3.21e5i)31-s + (1.09e6 − 6.34e5i)33-s + (1.45e6 + 2.52e6i)37-s + ⋯
L(s)  = 1  + (1.17 + 0.679i)3-s + (0.0334 − 0.0193i)5-s + (0.423 + 0.733i)9-s + (0.393 − 0.681i)11-s + 0.566i·13-s + 0.0524·15-s + (0.141 + 0.0817i)17-s + (0.734 − 0.423i)19-s + (0.170 + 0.295i)23-s + (−0.499 + 0.864i)25-s − 0.208i·27-s + 1.36·29-s + (−0.602 − 0.348i)31-s + (0.926 − 0.535i)33-s + (0.777 + 1.34i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.611 - 0.791i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.611 - 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $0.611 - 0.791i$
Analytic conductor: \(79.8462\)
Root analytic conductor: \(8.93567\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{196} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 196,\ (\ :4),\ 0.611 - 0.791i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(3.651596773\)
\(L(\frac12)\) \(\approx\) \(3.651596773\)
\(L(5)\) 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 \( 1 + (-95.3 - 55.0i)T + (3.28e3 + 5.68e3i)T^{2} \)
5 \( 1 + (-20.9 + 12.0i)T + (1.95e5 - 3.38e5i)T^{2} \)
11 \( 1 + (-5.76e3 + 9.98e3i)T + (-1.07e8 - 1.85e8i)T^{2} \)
13 \( 1 - 1.61e4iT - 8.15e8T^{2} \)
17 \( 1 + (-1.18e4 - 6.82e3i)T + (3.48e9 + 6.04e9i)T^{2} \)
19 \( 1 + (-9.56e4 + 5.52e4i)T + (8.49e9 - 1.47e10i)T^{2} \)
23 \( 1 + (-4.77e4 - 8.27e4i)T + (-3.91e10 + 6.78e10i)T^{2} \)
29 \( 1 - 9.68e5T + 5.00e11T^{2} \)
31 \( 1 + (5.56e5 + 3.21e5i)T + (4.26e11 + 7.38e11i)T^{2} \)
37 \( 1 + (-1.45e6 - 2.52e6i)T + (-1.75e12 + 3.04e12i)T^{2} \)
41 \( 1 - 2.54e6iT - 7.98e12T^{2} \)
43 \( 1 - 4.32e6T + 1.16e13T^{2} \)
47 \( 1 + (-8.23e6 + 4.75e6i)T + (1.19e13 - 2.06e13i)T^{2} \)
53 \( 1 + (1.66e6 - 2.88e6i)T + (-3.11e13 - 5.39e13i)T^{2} \)
59 \( 1 + (-1.22e6 - 7.08e5i)T + (7.34e13 + 1.27e14i)T^{2} \)
61 \( 1 + (9.79e6 - 5.65e6i)T + (9.58e13 - 1.66e14i)T^{2} \)
67 \( 1 + (3.61e6 - 6.26e6i)T + (-2.03e14 - 3.51e14i)T^{2} \)
71 \( 1 + 4.54e6T + 6.45e14T^{2} \)
73 \( 1 + (-1.47e7 - 8.50e6i)T + (4.03e14 + 6.98e14i)T^{2} \)
79 \( 1 + (2.65e7 + 4.60e7i)T + (-7.58e14 + 1.31e15i)T^{2} \)
83 \( 1 - 4.31e7iT - 2.25e15T^{2} \)
89 \( 1 + (6.63e7 - 3.83e7i)T + (1.96e15 - 3.40e15i)T^{2} \)
97 \( 1 + 2.32e7iT - 7.83e15T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.09922777377598401305339984341, −9.873600624052521802451455908155, −9.186964063120222568581390883209, −8.406165639768905744891285912245, −7.31489641643259010968362970646, −5.94487391126996026109136398897, −4.53123809444395206461769641437, −3.52352456996778642705460825124, −2.60901693469142635813043115864, −1.09393268988008470916480571616, 0.839050550823157389071018971074, 2.05946939873317604208986518044, 2.99818067432363001062624775783, 4.25468357831845652578214615038, 5.78946530403718406692802235102, 7.12993891587467866486179657904, 7.80960312949018712910797245047, 8.790362251547288010475779243662, 9.678203765203247554200358452804, 10.78553968155856674339060694783

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