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Properties

Label 2-14e2-7.5-c8-0-16
Degree $2$
Conductor $196$
Sign $0.832 - 0.553i$
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  + (85.0 − 49.0i)3-s + (951. + 549. i)5-s + (1.54e3 − 2.66e3i)9-s + (8.37e3 + 1.45e4i)11-s − 1.21e4i·13-s + 1.07e5·15-s + (−9.20e4 + 5.31e4i)17-s + (3.38e4 + 1.95e4i)19-s + (2.17e5 − 3.76e5i)23-s + (4.08e5 + 7.07e5i)25-s + 3.41e5i·27-s + 7.52e5·29-s + (6.11e5 − 3.53e5i)31-s + (1.42e6 + 8.22e5i)33-s + (−1.25e6 + 2.17e6i)37-s + ⋯
L(s)  = 1  + (1.04 − 0.606i)3-s + (1.52 + 0.878i)5-s + (0.234 − 0.406i)9-s + (0.571 + 0.990i)11-s − 0.426i·13-s + 2.13·15-s + (−1.10 + 0.636i)17-s + (0.259 + 0.150i)19-s + (0.777 − 1.34i)23-s + (1.04 + 1.81i)25-s + 0.642i·27-s + 1.06·29-s + (0.662 − 0.382i)31-s + (1.20 + 0.693i)33-s + (−0.671 + 1.16i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $0.832 - 0.553i$
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: \((2,\ 196,\ (\ :4),\ 0.832 - 0.553i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(4.522459375\)
\(L(\frac12)\) \(\approx\) \(4.522459375\)
\(L(5)\) 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 \( 1 + (-85.0 + 49.0i)T + (3.28e3 - 5.68e3i)T^{2} \)
5 \( 1 + (-951. - 549. i)T + (1.95e5 + 3.38e5i)T^{2} \)
11 \( 1 + (-8.37e3 - 1.45e4i)T + (-1.07e8 + 1.85e8i)T^{2} \)
13 \( 1 + 1.21e4iT - 8.15e8T^{2} \)
17 \( 1 + (9.20e4 - 5.31e4i)T + (3.48e9 - 6.04e9i)T^{2} \)
19 \( 1 + (-3.38e4 - 1.95e4i)T + (8.49e9 + 1.47e10i)T^{2} \)
23 \( 1 + (-2.17e5 + 3.76e5i)T + (-3.91e10 - 6.78e10i)T^{2} \)
29 \( 1 - 7.52e5T + 5.00e11T^{2} \)
31 \( 1 + (-6.11e5 + 3.53e5i)T + (4.26e11 - 7.38e11i)T^{2} \)
37 \( 1 + (1.25e6 - 2.17e6i)T + (-1.75e12 - 3.04e12i)T^{2} \)
41 \( 1 - 3.11e6iT - 7.98e12T^{2} \)
43 \( 1 - 1.65e6T + 1.16e13T^{2} \)
47 \( 1 + (5.50e6 + 3.17e6i)T + (1.19e13 + 2.06e13i)T^{2} \)
53 \( 1 + (-2.79e6 - 4.83e6i)T + (-3.11e13 + 5.39e13i)T^{2} \)
59 \( 1 + (1.66e6 - 9.62e5i)T + (7.34e13 - 1.27e14i)T^{2} \)
61 \( 1 + (1.54e7 + 8.93e6i)T + (9.58e13 + 1.66e14i)T^{2} \)
67 \( 1 + (-5.09e5 - 8.82e5i)T + (-2.03e14 + 3.51e14i)T^{2} \)
71 \( 1 - 2.80e6T + 6.45e14T^{2} \)
73 \( 1 + (-1.40e7 + 8.13e6i)T + (4.03e14 - 6.98e14i)T^{2} \)
79 \( 1 + (-2.90e7 + 5.03e7i)T + (-7.58e14 - 1.31e15i)T^{2} \)
83 \( 1 - 6.45e7iT - 2.25e15T^{2} \)
89 \( 1 + (-7.07e7 - 4.08e7i)T + (1.96e15 + 3.40e15i)T^{2} \)
97 \( 1 + 9.95e7iT - 7.83e15T^{2} \)
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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

−10.84381763975282933834327443678, −10.03108225131709069584041361691, −9.141495365334713407927049480035, −8.165772324790379265236088221991, −6.83788448577369816227956718203, −6.37645319392393459508339826472, −4.77470429648566518916178646279, −3.02192988940767209442753051606, −2.27975900949082354770225852672, −1.42465102677019194426006188954, 0.899816008033436830350652509589, 2.09547902663972726843000117938, 3.20545188326502185666036562368, 4.53444978358545462443571951486, 5.59888289013724543381518285147, 6.72043069236567868129768109690, 8.424805285709654516988658419112, 9.198916384114912474461525522686, 9.421906647714248702423705401057, 10.69021296825603700923737131852

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