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Properties

Label 1-14e2-196.27-r0-0-0
Degree $1$
Conductor $196$
Sign $0.820 + 0.572i$
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  + (−0.900 − 0.433i)3-s + (0.900 + 0.433i)5-s + (0.623 + 0.781i)9-s + (−0.623 + 0.781i)11-s + (−0.623 + 0.781i)13-s + (−0.623 − 0.781i)15-s + (0.222 − 0.974i)17-s + 19-s + (0.222 + 0.974i)23-s + (0.623 + 0.781i)25-s + (−0.222 − 0.974i)27-s + (−0.222 + 0.974i)29-s + 31-s + (0.900 − 0.433i)33-s + (−0.222 + 0.974i)37-s + ⋯
L(s)  = 1  + (−0.900 − 0.433i)3-s + (0.900 + 0.433i)5-s + (0.623 + 0.781i)9-s + (−0.623 + 0.781i)11-s + (−0.623 + 0.781i)13-s + (−0.623 − 0.781i)15-s + (0.222 − 0.974i)17-s + 19-s + (0.222 + 0.974i)23-s + (0.623 + 0.781i)25-s + (−0.222 − 0.974i)27-s + (−0.222 + 0.974i)29-s + 31-s + (0.900 − 0.433i)33-s + (−0.222 + 0.974i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.820 + 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.820 + 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $0.820 + 0.572i$
Analytic conductor: \(0.910220\)
Root analytic conductor: \(0.910220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{196} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 196,\ (0:\ ),\ 0.820 + 0.572i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9121239538 + 0.2866988598i\)
\(L(\frac12)\) \(\approx\) \(0.9121239538 + 0.2866988598i\)
\(L(1)\) \(\approx\) \(0.9122114098 + 0.08938014412i\)
\(L(1)\) \(\approx\) \(0.9122114098 + 0.08938014412i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.900 - 0.433i)T \)
5 \( 1 + (0.900 + 0.433i)T \)
11 \( 1 + (-0.623 + 0.781i)T \)
13 \( 1 + (-0.623 + 0.781i)T \)
17 \( 1 + (0.222 - 0.974i)T \)
19 \( 1 + T \)
23 \( 1 + (0.222 + 0.974i)T \)
29 \( 1 + (-0.222 + 0.974i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.222 + 0.974i)T \)
41 \( 1 + (0.900 + 0.433i)T \)
43 \( 1 + (0.900 - 0.433i)T \)
47 \( 1 + (0.623 - 0.781i)T \)
53 \( 1 + (-0.222 - 0.974i)T \)
59 \( 1 + (-0.900 + 0.433i)T \)
61 \( 1 + (0.222 - 0.974i)T \)
67 \( 1 - T \)
71 \( 1 + (0.222 + 0.974i)T \)
73 \( 1 + (-0.623 - 0.781i)T \)
79 \( 1 - T \)
83 \( 1 + (0.623 + 0.781i)T \)
89 \( 1 + (-0.623 - 0.781i)T \)
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

−26.84811797788738172091162021744, −26.17668657007645911197602996425, −24.70643367864668666856808721147, −24.21583876233609140203521371893, −22.9311289000434060905983701227, −22.124897808580024197224195440523, −21.24424903654838254224685939974, −20.585476762979812815529852829550, −19.09820435919891333489077881054, −17.92701672374850390735913775400, −17.27425561537336144912936779980, −16.38349103790088343414201857003, −15.46187028790607888579990809050, −14.17758956603704212984370373009, −12.98177815077823613003403679813, −12.222968687688256149919128005693, −10.82113233060361425765739339184, −10.15636865552707154301318854298, −9.097088291169151948826858935945, −7.71700313872004105490883089215, −6.09123947499997653868773452473, −5.55272329849112759977250503567, −4.40607694485826190084433498591, −2.7497016074263956099757872558, −0.91828249415012512720552669038, 1.51065056205098865055646808298, 2.73998911569220433933148714978, 4.79545632748218377730712093478, 5.56718377100925058646232569807, 6.865235315177907179763696013493, 7.48513246342444829198140543925, 9.437314337536506131075561704, 10.163095615623999532157224162162, 11.344972583829317709753117871616, 12.254635745322213996204914622502, 13.3780373822185868845734106350, 14.17996582940981244318046695725, 15.56982937981147157680063948631, 16.65259647095856116192970760985, 17.605980427514805382686216947209, 18.22112021565898690039368548099, 19.12384502532813536297907824598, 20.5564763331765138820290100641, 21.57931477621760824848332260560, 22.38988416711795588214136987899, 23.18661549684982535595747709708, 24.23601841160183024186759551176, 25.10092819119759738074717312639, 26.055210233524526984593257268066, 27.11606029383870686015684368034

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