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Properties

Label 2-14e2-7.3-c8-0-18
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  + (80.8 + 46.6i)3-s + (−327. + 188. i)5-s + (1.07e3 + 1.86e3i)9-s + (−9.66e3 + 1.67e4i)11-s − 4.84e4i·13-s − 3.52e4·15-s + (−4.75e3 − 2.74e3i)17-s + (7.13e4 − 4.11e4i)19-s + (−9.98e4 − 1.72e5i)23-s + (−1.23e5 + 2.14e5i)25-s − 4.11e5i·27-s + 7.29e5·29-s + (6.57e5 + 3.79e5i)31-s + (−1.56e6 + 9.02e5i)33-s + (−1.20e6 − 2.09e6i)37-s + ⋯
L(s)  = 1  + (0.997 + 0.576i)3-s + (−0.523 + 0.302i)5-s + (0.163 + 0.283i)9-s + (−0.660 + 1.14i)11-s − 1.69i·13-s − 0.696·15-s + (−0.0569 − 0.0328i)17-s + (0.547 − 0.316i)19-s + (−0.356 − 0.617i)23-s + (−0.317 + 0.549i)25-s − 0.774i·27-s + 1.03·29-s + (0.712 + 0.411i)31-s + (−1.31 + 0.760i)33-s + (−0.645 − 1.11i)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: $\chi_{196} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 196,\ (\ :4),\ 0.832 + 0.553i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(2.181367727\)
\(L(\frac12)\) \(\approx\) \(2.181367727\)
\(L(5)\) 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 \( 1 + (-80.8 - 46.6i)T + (3.28e3 + 5.68e3i)T^{2} \)
5 \( 1 + (327. - 188. i)T + (1.95e5 - 3.38e5i)T^{2} \)
11 \( 1 + (9.66e3 - 1.67e4i)T + (-1.07e8 - 1.85e8i)T^{2} \)
13 \( 1 + 4.84e4iT - 8.15e8T^{2} \)
17 \( 1 + (4.75e3 + 2.74e3i)T + (3.48e9 + 6.04e9i)T^{2} \)
19 \( 1 + (-7.13e4 + 4.11e4i)T + (8.49e9 - 1.47e10i)T^{2} \)
23 \( 1 + (9.98e4 + 1.72e5i)T + (-3.91e10 + 6.78e10i)T^{2} \)
29 \( 1 - 7.29e5T + 5.00e11T^{2} \)
31 \( 1 + (-6.57e5 - 3.79e5i)T + (4.26e11 + 7.38e11i)T^{2} \)
37 \( 1 + (1.20e6 + 2.09e6i)T + (-1.75e12 + 3.04e12i)T^{2} \)
41 \( 1 + 2.65e6iT - 7.98e12T^{2} \)
43 \( 1 - 3.43e6T + 1.16e13T^{2} \)
47 \( 1 + (-3.34e6 + 1.93e6i)T + (1.19e13 - 2.06e13i)T^{2} \)
53 \( 1 + (1.43e6 - 2.48e6i)T + (-3.11e13 - 5.39e13i)T^{2} \)
59 \( 1 + (-1.34e7 - 7.77e6i)T + (7.34e13 + 1.27e14i)T^{2} \)
61 \( 1 + (-8.84e6 + 5.10e6i)T + (9.58e13 - 1.66e14i)T^{2} \)
67 \( 1 + (-3.64e6 + 6.31e6i)T + (-2.03e14 - 3.51e14i)T^{2} \)
71 \( 1 + 4.30e7T + 6.45e14T^{2} \)
73 \( 1 + (1.32e7 + 7.67e6i)T + (4.03e14 + 6.98e14i)T^{2} \)
79 \( 1 + (1.11e7 + 1.93e7i)T + (-7.58e14 + 1.31e15i)T^{2} \)
83 \( 1 - 8.50e7iT - 2.25e15T^{2} \)
89 \( 1 + (-2.69e7 + 1.55e7i)T + (1.96e15 - 3.40e15i)T^{2} \)
97 \( 1 + 9.56e7iT - 7.83e15T^{2} \)
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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

−10.56108155456074807968244384441, −10.05131901582119289596271752093, −8.889989982849337395469806847801, −7.958956735593963680376738546122, −7.19845794295535977747249950446, −5.53194446532166365144093028332, −4.32565565629027251965349936007, −3.22502661819602936601834543998, −2.41714387582926199713225153496, −0.49063268314647950024515263605, 1.04234942685475484593895361068, 2.29069577439730798640139895398, 3.37606273910892826367047291969, 4.58534781982184182282939549737, 6.08161543271214817702103422628, 7.30862239552642854457696005751, 8.226051695822019496147434474141, 8.772000264570716153470353506677, 9.969178889919125112630091135425, 11.35919809302778319145610143450

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