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Properties

Label 1-14e2-196.15-r1-0-0
Degree $1$
Conductor $196$
Sign $0.0320 - 0.999i$
Analytic cond. $21.0631$
Root an. cond. $21.0631$
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  + (0.222 + 0.974i)3-s + (−0.222 − 0.974i)5-s + (−0.900 + 0.433i)9-s + (0.900 + 0.433i)11-s + (−0.900 − 0.433i)13-s + (0.900 − 0.433i)15-s + (0.623 − 0.781i)17-s − 19-s + (−0.623 − 0.781i)23-s + (−0.900 + 0.433i)25-s + (−0.623 − 0.781i)27-s + (0.623 − 0.781i)29-s − 31-s + (−0.222 + 0.974i)33-s + (0.623 − 0.781i)37-s + ⋯
L(s)  = 1  + (0.222 + 0.974i)3-s + (−0.222 − 0.974i)5-s + (−0.900 + 0.433i)9-s + (0.900 + 0.433i)11-s + (−0.900 − 0.433i)13-s + (0.900 − 0.433i)15-s + (0.623 − 0.781i)17-s − 19-s + (−0.623 − 0.781i)23-s + (−0.900 + 0.433i)25-s + (−0.623 − 0.781i)27-s + (0.623 − 0.781i)29-s − 31-s + (−0.222 + 0.974i)33-s + (0.623 − 0.781i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0320 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0320 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $0.0320 - 0.999i$
Analytic conductor: \(21.0631\)
Root analytic conductor: \(21.0631\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{196} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 196,\ (1:\ ),\ 0.0320 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7639471199 - 0.7398415281i\)
\(L(\frac12)\) \(\approx\) \(0.7639471199 - 0.7398415281i\)
\(L(1)\) \(\approx\) \(0.9458814049 + 0.02511800747i\)
\(L(1)\) \(\approx\) \(0.9458814049 + 0.02511800747i\)

Euler product

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

−26.89942281544891363053895392440, −25.84059688548309949976709082835, −25.22767119341870978223035806744, −23.97593558798327128163092590390, −23.44629858851963301830730942921, −22.22914184108165882165133511302, −21.50089678893082695022285179904, −19.762507007734058517608632530207, −19.408493172568692643101698327910, −18.47266522065953139628512909705, −17.50097954627331636218927277593, −16.55119815316824504722775650817, −14.77231605108332956970748532526, −14.529797604090277447606296579644, −13.31713915841279976283701739743, −12.1309156184104163080103079522, −11.403561112455636974273482513516, −10.1208750199834887663059319267, −8.77306519591093918127537670924, −7.66428135818820522763236033070, −6.757815177086381190891542365689, −5.90252534011268796210590217387, −3.949658006865311010893229965916, −2.77758250890859630284185744323, −1.515028309895339630602242430664, 0.35090145303666559033355616527, 2.29471667601366701627696372666, 3.90364913086243538829542984407, 4.68085281495467483653604439315, 5.77567844642219149193270900988, 7.494724167571243537244897430535, 8.67080323284830556359229126455, 9.46470253633031354967601355062, 10.41904929433207296487221469278, 11.80385449121003794271577580830, 12.56931926815777159191361348312, 14.0401791673425832098270692842, 14.88908907132379169045359369533, 15.901820432832888145870605586779, 16.7962981439818275873796576794, 17.44547114828692549642679670394, 19.16762405068996700308858898908, 20.08404737507163829997483982917, 20.64727900287939948659016476462, 21.736621810971199797189937888615, 22.57519507152124780802744986787, 23.62295301200006491186196149570, 24.90418768605716299152971245240, 25.393092871983829198326253092996, 26.77110430004551715462881467795

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