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Properties

Label 2-14e2-7.3-c8-0-6
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (127. + 73.4i)3-s + (−834. + 482. i)5-s + (7.49e3 + 1.29e4i)9-s + (−5.92e3 + 1.02e4i)11-s + 2.88e4i·13-s − 1.41e5·15-s + (3.98e4 + 2.30e4i)17-s + (−1.68e4 + 9.74e3i)19-s + (6.58e4 + 1.14e5i)23-s + (2.69e5 − 4.66e5i)25-s + 1.23e6i·27-s + 1.22e6·29-s + (−9.41e5 − 5.43e5i)31-s + (−1.50e6 + 8.70e5i)33-s + (−1.24e6 − 2.15e6i)37-s + ⋯
L(s)  = 1  + (1.56 + 0.906i)3-s + (−1.33 + 0.771i)5-s + (1.14 + 1.97i)9-s + (−0.404 + 0.701i)11-s + 1.01i·13-s − 2.79·15-s + (0.477 + 0.275i)17-s + (−0.129 + 0.0747i)19-s + (0.235 + 0.407i)23-s + (0.689 − 1.19i)25-s + 2.33i·27-s + 1.73·29-s + (−1.01 − 0.588i)31-s + (−1.27 + 0.733i)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: $\chi_{196} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 196,\ (\ :4),\ -0.895 + 0.444i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.899512481\)
\(L(\frac12)\) \(\approx\) \(1.899512481\)
\(L(5)\) 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 \( 1 + (-127. - 73.4i)T + (3.28e3 + 5.68e3i)T^{2} \)
5 \( 1 + (834. - 482. i)T + (1.95e5 - 3.38e5i)T^{2} \)
11 \( 1 + (5.92e3 - 1.02e4i)T + (-1.07e8 - 1.85e8i)T^{2} \)
13 \( 1 - 2.88e4iT - 8.15e8T^{2} \)
17 \( 1 + (-3.98e4 - 2.30e4i)T + (3.48e9 + 6.04e9i)T^{2} \)
19 \( 1 + (1.68e4 - 9.74e3i)T + (8.49e9 - 1.47e10i)T^{2} \)
23 \( 1 + (-6.58e4 - 1.14e5i)T + (-3.91e10 + 6.78e10i)T^{2} \)
29 \( 1 - 1.22e6T + 5.00e11T^{2} \)
31 \( 1 + (9.41e5 + 5.43e5i)T + (4.26e11 + 7.38e11i)T^{2} \)
37 \( 1 + (1.24e6 + 2.15e6i)T + (-1.75e12 + 3.04e12i)T^{2} \)
41 \( 1 + 2.64e4iT - 7.98e12T^{2} \)
43 \( 1 + 4.08e6T + 1.16e13T^{2} \)
47 \( 1 + (6.01e6 - 3.47e6i)T + (1.19e13 - 2.06e13i)T^{2} \)
53 \( 1 + (-4.38e6 + 7.60e6i)T + (-3.11e13 - 5.39e13i)T^{2} \)
59 \( 1 + (3.24e6 + 1.87e6i)T + (7.34e13 + 1.27e14i)T^{2} \)
61 \( 1 + (1.21e7 - 7.04e6i)T + (9.58e13 - 1.66e14i)T^{2} \)
67 \( 1 + (-3.81e6 + 6.60e6i)T + (-2.03e14 - 3.51e14i)T^{2} \)
71 \( 1 - 1.58e5T + 6.45e14T^{2} \)
73 \( 1 + (-3.36e7 - 1.94e7i)T + (4.03e14 + 6.98e14i)T^{2} \)
79 \( 1 + (5.65e6 + 9.78e6i)T + (-7.58e14 + 1.31e15i)T^{2} \)
83 \( 1 - 5.38e6iT - 2.25e15T^{2} \)
89 \( 1 + (-9.42e6 + 5.44e6i)T + (1.96e15 - 3.40e15i)T^{2} \)
97 \( 1 - 5.55e7iT - 7.83e15T^{2} \)
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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

−11.39279150586945522006645919693, −10.43305938696992851419798706833, −9.587661479309392725317472281553, −8.525902599582850792064688683130, −7.75831385006525724296727212086, −6.94492116219395541183715174486, −4.76320078626914744767846891301, −3.88617552650721471634663761869, −3.15175903308057942562559138071, −2.00187449403573613196155870763, 0.35607688800255573412800756851, 1.28262048495552276208410534551, 2.90894419637170520447694415457, 3.53429140922431557623953180897, 4.94979162446139006160845916391, 6.74078668982039940534310101396, 7.85157248568691278409469360420, 8.249431144474212240161331208317, 8.938403044176295755427446036520, 10.35161519226440353030888705283

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