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Properties

Label 2-14e2-7.3-c8-0-25
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  + (29.0 + 16.7i)3-s + (288. − 166. i)5-s + (−2.71e3 − 4.70e3i)9-s + (1.20e4 − 2.08e4i)11-s − 4.30e4i·13-s + 1.12e4·15-s + (4.84e4 + 2.79e4i)17-s + (−1.80e5 + 1.04e5i)19-s + (7.98e4 + 1.38e5i)23-s + (−1.39e5 + 2.41e5i)25-s − 4.02e5i·27-s − 3.26e5·29-s + (−1.13e6 − 6.52e5i)31-s + (7.01e5 − 4.05e5i)33-s + (−7.97e5 − 1.38e6i)37-s + ⋯
L(s)  = 1  + (0.359 + 0.207i)3-s + (0.462 − 0.266i)5-s + (−0.414 − 0.717i)9-s + (0.823 − 1.42i)11-s − 1.50i·13-s + 0.221·15-s + (0.580 + 0.335i)17-s + (−1.38 + 0.798i)19-s + (0.285 + 0.494i)23-s + (−0.357 + 0.619i)25-s − 0.757i·27-s − 0.461·29-s + (−1.22 − 0.706i)31-s + (0.591 − 0.341i)33-s + (−0.425 − 0.736i)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

\(L(\frac{9}{2})\) \(\approx\) \(1.320514318\)
\(L(\frac12)\) \(\approx\) \(1.320514318\)
\(L(5)\) 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 \( 1 + (-29.0 - 16.7i)T + (3.28e3 + 5.68e3i)T^{2} \)
5 \( 1 + (-288. + 166. i)T + (1.95e5 - 3.38e5i)T^{2} \)
11 \( 1 + (-1.20e4 + 2.08e4i)T + (-1.07e8 - 1.85e8i)T^{2} \)
13 \( 1 + 4.30e4iT - 8.15e8T^{2} \)
17 \( 1 + (-4.84e4 - 2.79e4i)T + (3.48e9 + 6.04e9i)T^{2} \)
19 \( 1 + (1.80e5 - 1.04e5i)T + (8.49e9 - 1.47e10i)T^{2} \)
23 \( 1 + (-7.98e4 - 1.38e5i)T + (-3.91e10 + 6.78e10i)T^{2} \)
29 \( 1 + 3.26e5T + 5.00e11T^{2} \)
31 \( 1 + (1.13e6 + 6.52e5i)T + (4.26e11 + 7.38e11i)T^{2} \)
37 \( 1 + (7.97e5 + 1.38e6i)T + (-1.75e12 + 3.04e12i)T^{2} \)
41 \( 1 - 4.17e6iT - 7.98e12T^{2} \)
43 \( 1 - 1.46e6T + 1.16e13T^{2} \)
47 \( 1 + (3.56e6 - 2.05e6i)T + (1.19e13 - 2.06e13i)T^{2} \)
53 \( 1 + (3.28e6 - 5.68e6i)T + (-3.11e13 - 5.39e13i)T^{2} \)
59 \( 1 + (-1.10e7 - 6.39e6i)T + (7.34e13 + 1.27e14i)T^{2} \)
61 \( 1 + (-1.43e6 + 8.29e5i)T + (9.58e13 - 1.66e14i)T^{2} \)
67 \( 1 + (-5.43e6 + 9.41e6i)T + (-2.03e14 - 3.51e14i)T^{2} \)
71 \( 1 + 2.55e7T + 6.45e14T^{2} \)
73 \( 1 + (1.55e7 + 8.97e6i)T + (4.03e14 + 6.98e14i)T^{2} \)
79 \( 1 + (2.66e7 + 4.60e7i)T + (-7.58e14 + 1.31e15i)T^{2} \)
83 \( 1 - 2.08e6iT - 2.25e15T^{2} \)
89 \( 1 + (-8.56e7 + 4.94e7i)T + (1.96e15 - 3.40e15i)T^{2} \)
97 \( 1 + 2.44e7iT - 7.83e15T^{2} \)
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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

−10.57587328973211814844233809255, −9.466870241492679200620342466515, −8.696298107928928417697865795469, −7.80637587067762183236593086966, −6.09014128279864153020929826048, −5.67399849681518785767758374081, −3.84245895669668845695275647641, −3.10399130918681980885543636608, −1.45762042680057382503361234596, −0.26640221562369518468088270588, 1.75125510029765069875147686131, 2.32879011134178333398378625335, 4.01805148560082171141840009663, 5.06770800489876991947542959652, 6.60041778614980332485538400718, 7.18251670770461297941932202452, 8.607689863755492788097314165179, 9.371723102685498254317564584187, 10.40708668285490450239898605553, 11.45713251124555998376558875496

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