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Properties

Label 2-14e2-7.2-c9-0-3
Degree $2$
Conductor $196$
Sign $0.386 + 0.922i$
Analytic cond. $100.947$
Root an. cond. $10.0472$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (123. + 213. i)3-s + (−1.00e3 + 1.74e3i)5-s + (−2.06e4 + 3.58e4i)9-s + (2.39e4 + 4.15e4i)11-s − 1.71e5·13-s − 4.97e5·15-s + (−1.67e5 − 2.89e5i)17-s + (9.63e3 − 1.66e4i)19-s + (6.62e5 − 1.14e6i)23-s + (−1.05e6 − 1.82e6i)25-s − 5.35e6·27-s − 1.11e6·29-s + (4.65e6 + 8.05e6i)31-s + (−5.92e6 + 1.02e7i)33-s + (8.46e6 − 1.46e7i)37-s + ⋯
L(s)  = 1  + (0.880 + 1.52i)3-s + (−0.720 + 1.24i)5-s + (−1.05 + 1.81i)9-s + (0.494 + 0.855i)11-s − 1.66·13-s − 2.53·15-s + (−0.486 − 0.842i)17-s + (0.0169 − 0.0293i)19-s + (0.493 − 0.855i)23-s + (−0.538 − 0.931i)25-s − 1.93·27-s − 0.294·29-s + (0.904 + 1.56i)31-s + (−0.870 + 1.50i)33-s + (0.742 − 1.28i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $0.386 + 0.922i$
Analytic conductor: \(100.947\)
Root analytic conductor: \(10.0472\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{196} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 196,\ (\ :9/2),\ 0.386 + 0.922i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.7692066193\)
\(L(\frac12)\) \(\approx\) \(0.7692066193\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-123. - 213. i)T + (-9.84e3 + 1.70e4i)T^{2} \)
5 \( 1 + (1.00e3 - 1.74e3i)T + (-9.76e5 - 1.69e6i)T^{2} \)
11 \( 1 + (-2.39e4 - 4.15e4i)T + (-1.17e9 + 2.04e9i)T^{2} \)
13 \( 1 + 1.71e5T + 1.06e10T^{2} \)
17 \( 1 + (1.67e5 + 2.89e5i)T + (-5.92e10 + 1.02e11i)T^{2} \)
19 \( 1 + (-9.63e3 + 1.66e4i)T + (-1.61e11 - 2.79e11i)T^{2} \)
23 \( 1 + (-6.62e5 + 1.14e6i)T + (-9.00e11 - 1.55e12i)T^{2} \)
29 \( 1 + 1.11e6T + 1.45e13T^{2} \)
31 \( 1 + (-4.65e6 - 8.05e6i)T + (-1.32e13 + 2.28e13i)T^{2} \)
37 \( 1 + (-8.46e6 + 1.46e7i)T + (-6.49e13 - 1.12e14i)T^{2} \)
41 \( 1 + 9.85e5T + 3.27e14T^{2} \)
43 \( 1 + 1.62e7T + 5.02e14T^{2} \)
47 \( 1 + (1.34e7 - 2.33e7i)T + (-5.59e14 - 9.69e14i)T^{2} \)
53 \( 1 + (-4.12e7 - 7.13e7i)T + (-1.64e15 + 2.85e15i)T^{2} \)
59 \( 1 + (6.53e7 + 1.13e8i)T + (-4.33e15 + 7.50e15i)T^{2} \)
61 \( 1 + (4.79e7 - 8.30e7i)T + (-5.84e15 - 1.01e16i)T^{2} \)
67 \( 1 + (-5.51e6 - 9.54e6i)T + (-1.36e16 + 2.35e16i)T^{2} \)
71 \( 1 + 2.35e8T + 4.58e16T^{2} \)
73 \( 1 + (-1.60e8 - 2.77e8i)T + (-2.94e16 + 5.09e16i)T^{2} \)
79 \( 1 + (1.94e7 - 3.36e7i)T + (-5.99e16 - 1.03e17i)T^{2} \)
83 \( 1 + 2.33e8T + 1.86e17T^{2} \)
89 \( 1 + (2.66e8 - 4.60e8i)T + (-1.75e17 - 3.03e17i)T^{2} \)
97 \( 1 - 1.18e9T + 7.60e17T^{2} \)
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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

−11.33013187351383993166743538934, −10.45288333785801894465358695085, −9.764031270602597633837957832520, −8.916602292465437199891936658417, −7.61687092753286224208126126412, −6.86957999845476209064831629090, −4.89914050243753945084506928969, −4.25180138341585079215582163097, −3.04363284729667091802009263212, −2.46103776749086130816659332607, 0.15773332746876120825701767414, 1.01836165074499454080283425226, 2.06231452645792670577794747563, 3.32720881848959026979778683106, 4.64437738079030316570690161531, 6.10144427174257368428676648423, 7.26931681675429000916463965281, 8.068248721355605772075326206861, 8.671606128676876500306247933658, 9.636368210379695784061456709852

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