Properties

Label 1-14e2-196.115-r0-0-0
Degree $1$
Conductor $196$
Sign $0.0853 + 0.996i$
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  + (0.826 + 0.563i)3-s + (−0.0747 + 0.997i)5-s + (0.365 + 0.930i)9-s + (−0.365 + 0.930i)11-s + (−0.623 − 0.781i)13-s + (−0.623 + 0.781i)15-s + (0.733 − 0.680i)17-s + (−0.5 + 0.866i)19-s + (0.733 + 0.680i)23-s + (−0.988 − 0.149i)25-s + (−0.222 + 0.974i)27-s + (−0.222 − 0.974i)29-s + (−0.5 − 0.866i)31-s + (−0.826 + 0.563i)33-s + (0.955 + 0.294i)37-s + ⋯
L(s)  = 1  + (0.826 + 0.563i)3-s + (−0.0747 + 0.997i)5-s + (0.365 + 0.930i)9-s + (−0.365 + 0.930i)11-s + (−0.623 − 0.781i)13-s + (−0.623 + 0.781i)15-s + (0.733 − 0.680i)17-s + (−0.5 + 0.866i)19-s + (0.733 + 0.680i)23-s + (−0.988 − 0.149i)25-s + (−0.222 + 0.974i)27-s + (−0.222 − 0.974i)29-s + (−0.5 − 0.866i)31-s + (−0.826 + 0.563i)33-s + (0.955 + 0.294i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0853 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0853 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $0.0853 + 0.996i$
Analytic conductor: \(0.910220\)
Root analytic conductor: \(0.910220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{196} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 196,\ (0:\ ),\ 0.0853 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.041332091 + 0.9559123325i\)
\(L(\frac12)\) \(\approx\) \(1.041332091 + 0.9559123325i\)
\(L(1)\) \(\approx\) \(1.167413243 + 0.5285438091i\)
\(L(1)\) \(\approx\) \(1.167413243 + 0.5285438091i\)

Euler product

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

−26.617518379777372949198068846409, −25.771868309864118391706683063445, −24.735042487093508610897110150041, −24.05134158396226823254677736015, −23.4301015168894300123189349055, −21.586672291697533680660863855698, −21.102942328307053208195466740022, −19.86736027763723624720879980228, −19.319884680969710093666177236492, −18.29440446194650092019103764846, −17.01482026737193581468113208307, −16.199519899384415603663623016371, −14.92152051353593554215169840086, −14.010365038877654929469032135269, −12.94317095520298851086848477392, −12.36385760759553892683878593797, −10.97451057937094304647834785668, −9.404218799633301910340613789164, −8.70067107523607979344560042311, −7.77991076739056744308086289670, −6.553581974218002069810260413469, −5.1365436383589330107176198677, −3.83619443043884884764228889070, −2.47720477357272647479613985337, −1.07319336895501779866438085521, 2.21657010887359320729266717720, 3.12679397794384910870974006378, 4.33153321768696263933140737158, 5.678256411543857347269132212680, 7.37675861059654528104820056309, 7.85560077102010181420012929275, 9.55261151884494464766592057713, 10.0982195236704398039348343510, 11.18110234018780794268442249487, 12.59633424782188132942427694386, 13.73195935449858487324889047232, 14.92237170435298021657886681831, 15.083262869501256323010921406997, 16.42615594087864787402564424314, 17.69717340038035607673677088781, 18.74406775959241196737067921986, 19.55911624829909076845368725069, 20.61170159058221045395269363694, 21.36913878654114960614377265292, 22.567434440876653564122225014, 23.06908622731035619135317909305, 24.69195963582888087730864753123, 25.54040062655478531786539253646, 26.13200923963474314009749539476, 27.28528066201269812057855222181

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