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Properties

Label 2-14e2-28.11-c2-0-12
Degree $2$
Conductor $196$
Sign $-0.605 - 0.795i$
Analytic cond. $5.34061$
Root an. cond. $2.31097$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + 1.73i)2-s + (3.87 + 2.23i)3-s + (−1.99 − 3.46i)4-s + (3.87 + 6.70i)5-s + (−7.74 + 4.47i)6-s + 7.99·8-s + (5.5 + 9.52i)9-s − 15.4·10-s + (−6 − 3.46i)11-s − 17.8i·12-s + 7.74·13-s + 34.6i·15-s + (−8 + 13.8i)16-s + (7.74 − 13.4i)17-s − 22·18-s + (−11.6 + 6.70i)19-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (1.29 + 0.745i)3-s + (−0.499 − 0.866i)4-s + (0.774 + 1.34i)5-s + (−1.29 + 0.745i)6-s + 0.999·8-s + (0.611 + 1.05i)9-s − 1.54·10-s + (−0.545 − 0.314i)11-s − 1.49i·12-s + 0.595·13-s + 2.30i·15-s + (−0.5 + 0.866i)16-s + (0.455 − 0.789i)17-s − 1.22·18-s + (−0.611 + 0.353i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $-0.605 - 0.795i$
Analytic conductor: \(5.34061\)
Root analytic conductor: \(2.31097\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{196} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 196,\ (\ :1),\ -0.605 - 0.795i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.834796 + 1.68398i\)
\(L(\frac12)\) \(\approx\) \(0.834796 + 1.68398i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - 1.73i)T \)
7 \( 1 \)
good3 \( 1 + (-3.87 - 2.23i)T + (4.5 + 7.79i)T^{2} \)
5 \( 1 + (-3.87 - 6.70i)T + (-12.5 + 21.6i)T^{2} \)
11 \( 1 + (6 + 3.46i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 - 7.74T + 169T^{2} \)
17 \( 1 + (-7.74 + 13.4i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (11.6 - 6.70i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-18 + 10.3i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + 26T + 841T^{2} \)
31 \( 1 + (23.2 + 13.4i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-1 - 1.73i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 46.4T + 1.68e3T^{2} \)
43 \( 1 - 48.4iT - 1.84e3T^{2} \)
47 \( 1 + (-69.7 + 40.2i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-23 + 39.8i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-58.0 - 33.5i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-3.87 - 6.70i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-78 - 45.0i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 96.9iT - 5.04e3T^{2} \)
73 \( 1 + (-61.9 + 107. i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (24 - 13.8i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 93.9iT - 6.88e3T^{2} \)
89 \( 1 + (-30.9 - 53.6i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 46.4T + 9.40e3T^{2} \)
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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

−13.31621316932641880676887472677, −11.01444763093290045768559516091, −10.30275621639599974264530695213, −9.545680329810204679648560391115, −8.670688119213508556519762001114, −7.64636462349319214981604809989, −6.59113086856440540252305797819, −5.38401036232956939965812514148, −3.67174098998013131668141367718, −2.37769750541695302595260581354, 1.29463947658377901422007707860, 2.26017160821249859874655044950, 3.77068029464125498731608309942, 5.32998524729954065184495064914, 7.24735769548457291896660352094, 8.357161248840728649428074313448, 8.832084874422845968191271028994, 9.639299546508971101579973712331, 10.80813950862713925326320130996, 12.31980856322038992991114562679

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