Properties

Label 1-14e2-196.131-r0-0-0
Degree $1$
Conductor $196$
Sign $-0.998 + 0.0534i$
Analytic cond. $0.910220$
Root an. cond. $0.910220$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.988 + 0.149i)3-s + (−0.365 + 0.930i)5-s + (0.955 − 0.294i)9-s + (−0.955 − 0.294i)11-s + (0.222 + 0.974i)13-s + (0.222 − 0.974i)15-s + (−0.826 + 0.563i)17-s + (−0.5 − 0.866i)19-s + (−0.826 − 0.563i)23-s + (−0.733 − 0.680i)25-s + (−0.900 + 0.433i)27-s + (−0.900 − 0.433i)29-s + (−0.5 + 0.866i)31-s + (0.988 + 0.149i)33-s + (0.0747 + 0.997i)37-s + ⋯
L(s)  = 1  + (−0.988 + 0.149i)3-s + (−0.365 + 0.930i)5-s + (0.955 − 0.294i)9-s + (−0.955 − 0.294i)11-s + (0.222 + 0.974i)13-s + (0.222 − 0.974i)15-s + (−0.826 + 0.563i)17-s + (−0.5 − 0.866i)19-s + (−0.826 − 0.563i)23-s + (−0.733 − 0.680i)25-s + (−0.900 + 0.433i)27-s + (−0.900 − 0.433i)29-s + (−0.5 + 0.866i)31-s + (0.988 + 0.149i)33-s + (0.0747 + 0.997i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0534i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0534i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $-0.998 + 0.0534i$
Analytic conductor: \(0.910220\)
Root analytic conductor: \(0.910220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{196} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 196,\ (0:\ ),\ -0.998 + 0.0534i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.005994641967 + 0.2243455038i\)
\(L(\frac12)\) \(\approx\) \(0.005994641967 + 0.2243455038i\)
\(L(1)\) \(\approx\) \(0.5098212792 + 0.1613897778i\)
\(L(1)\) \(\approx\) \(0.5098212792 + 0.1613897778i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.988 + 0.149i)T \)
5 \( 1 + (-0.365 + 0.930i)T \)
11 \( 1 + (-0.955 - 0.294i)T \)
13 \( 1 + (0.222 + 0.974i)T \)
17 \( 1 + (-0.826 + 0.563i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.826 - 0.563i)T \)
29 \( 1 + (-0.900 - 0.433i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (0.0747 + 0.997i)T \)
41 \( 1 + (-0.623 - 0.781i)T \)
43 \( 1 + (-0.623 + 0.781i)T \)
47 \( 1 + (-0.733 + 0.680i)T \)
53 \( 1 + (0.0747 - 0.997i)T \)
59 \( 1 + (0.365 + 0.930i)T \)
61 \( 1 + (-0.0747 - 0.997i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (0.900 - 0.433i)T \)
73 \( 1 + (0.733 + 0.680i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.222 + 0.974i)T \)
89 \( 1 + (-0.955 + 0.294i)T \)
97 \( 1 - T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−26.76163124173248565748839627600, −25.36175012847996017330952731643, −24.45545993822531368351865810421, −23.59808232755224980120399650737, −22.936623357278995639818443723860, −21.86716056905524327828509749225, −20.74345901203015240172676814210, −19.99594920652931208135201687392, −18.58189876119825660970833180278, −17.847949518972428900445199370300, −16.81197780035227245400259911497, −15.982826982081798048910541324559, −15.20468324272108564979405461530, −13.31746126246358168047704647397, −12.77308028484442636621790892900, −11.76586582084285442043249561400, −10.750606695974172207128946236987, −9.69675446961219190893091152676, −8.23680928658317969466154190686, −7.348811610018435165202718393870, −5.81643772783084132865647554774, −5.08634614723198041959537169409, −3.89788261003620480628227180206, −1.87062876889792794635345559520, −0.18614746575628735811303144647, 2.13141587856808806566817080393, 3.74293093104868047618454701753, 4.867733579166809269181094286044, 6.26709695197275186834338014027, 6.92884755007289278019490868011, 8.29318684986910678507904488597, 9.83226621899645772229087361472, 10.89742734082139856250800744940, 11.34911853213986710496685408023, 12.61993963394226464209555427161, 13.72795836047823486555840158859, 15.08378767923242466026130383742, 15.81890388411113985928658371212, 16.818344035013922354859529011088, 17.99041483527288964605985613326, 18.58616768736488448363415746574, 19.61812837864569270067926091384, 21.11722474823743829403507310236, 21.891104426115869447527381042430, 22.65229408928702782762095563448, 23.768830125035438502623129851008, 24.08309926791939555600666922693, 25.965492604343645241329821900533, 26.44439023530781582820755561179, 27.45274059382133107646647017673

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