Properties

Label 2-14e2-7.3-c4-0-11
Degree $2$
Conductor $196$
Sign $-0.895 + 0.444i$
Analytic cond. $20.2605$
Root an. cond. $4.50116$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (6 + 3.46i)3-s + (−18 + 10.3i)5-s + (−16.5 − 28.5i)9-s + (−9 + 15.5i)11-s + 131. i·13-s − 144·15-s + (−360 − 207. i)17-s + (−78 + 45.0i)19-s + (−369 − 639. i)23-s + (−96.5 + 167. i)25-s − 789. i·27-s − 846·29-s + (1.00e3 + 581. i)31-s + (−108 + 62.3i)33-s + (−1.19e3 − 2.06e3i)37-s + ⋯
L(s)  = 1  + (0.666 + 0.384i)3-s + (−0.719 + 0.415i)5-s + (−0.203 − 0.352i)9-s + (−0.0743 + 0.128i)11-s + 0.778i·13-s − 0.640·15-s + (−1.24 − 0.719i)17-s + (−0.216 + 0.124i)19-s + (−0.697 − 1.20i)23-s + (−0.154 + 0.267i)25-s − 1.08i·27-s − 1.00·29-s + (1.04 + 0.605i)31-s + (−0.0991 + 0.0572i)33-s + (−0.871 − 1.50i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $-0.895 + 0.444i$
Analytic conductor: \(20.2605\)
Root analytic conductor: \(4.50116\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{196} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 196,\ (\ :2),\ -0.895 + 0.444i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.01263888894\)
\(L(\frac12)\) \(\approx\) \(0.01263888894\)
\(L(3)\) 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 \( 1 + (-6 - 3.46i)T + (40.5 + 70.1i)T^{2} \)
5 \( 1 + (18 - 10.3i)T + (312.5 - 541. i)T^{2} \)
11 \( 1 + (9 - 15.5i)T + (-7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 - 131. iT - 2.85e4T^{2} \)
17 \( 1 + (360 + 207. i)T + (4.17e4 + 7.23e4i)T^{2} \)
19 \( 1 + (78 - 45.0i)T + (6.51e4 - 1.12e5i)T^{2} \)
23 \( 1 + (369 + 639. i)T + (-1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + 846T + 7.07e5T^{2} \)
31 \( 1 + (-1.00e3 - 581. i)T + (4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 + (1.19e3 + 2.06e3i)T + (-9.37e5 + 1.62e6i)T^{2} \)
41 \( 1 - 1.70e3iT - 2.82e6T^{2} \)
43 \( 1 + 2.51e3T + 3.41e6T^{2} \)
47 \( 1 + (-2.95e3 + 1.70e3i)T + (2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (-135 + 233. i)T + (-3.94e6 - 6.83e6i)T^{2} \)
59 \( 1 + (2.71e3 + 1.56e3i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (5.62e3 - 3.24e3i)T + (6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (1.22e3 - 2.12e3i)T + (-1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 + 3.15e3T + 2.54e7T^{2} \)
73 \( 1 + (-204 - 117. i)T + (1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (-1.99e3 - 3.44e3i)T + (-1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 - 5.00e3iT - 4.74e7T^{2} \)
89 \( 1 + (-6.58e3 + 3.80e3i)T + (3.13e7 - 5.43e7i)T^{2} \)
97 \( 1 - 1.25e4iT - 8.85e7T^{2} \)
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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

−11.45031406501843444833230755409, −10.41534168773278239848272853770, −9.239110249289908287923264769890, −8.548586476900154399449956689309, −7.33625907554476744211347534705, −6.34100930860520269385850789409, −4.56849704977143777620494654009, −3.62343190327263188075951768760, −2.33347595533821223793800754881, −0.00384578003268010108212208163, 1.87684391567690858230229859132, 3.31942888513962223752717895503, 4.60784986466549772076417942705, 6.00515402687266773175453579706, 7.48298301216332544895968464565, 8.183034070337410344701333947566, 8.941403638124861725205268795774, 10.32987121873340984334414566179, 11.34365450536848832177894573131, 12.31258156806841794569640000463

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