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Properties

Label 1-14e2-196.51-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  + (0.733 + 0.680i)3-s + (0.955 − 0.294i)5-s + (0.0747 + 0.997i)9-s + (−0.0747 + 0.997i)11-s + (−0.900 + 0.433i)13-s + (0.900 + 0.433i)15-s + (−0.988 + 0.149i)17-s + (0.5 + 0.866i)19-s + (0.988 + 0.149i)23-s + (0.826 − 0.563i)25-s + (−0.623 + 0.781i)27-s + (0.623 + 0.781i)29-s + (0.5 − 0.866i)31-s + (−0.733 + 0.680i)33-s + (0.365 − 0.930i)37-s + ⋯
L(s)  = 1  + (0.733 + 0.680i)3-s + (0.955 − 0.294i)5-s + (0.0747 + 0.997i)9-s + (−0.0747 + 0.997i)11-s + (−0.900 + 0.433i)13-s + (0.900 + 0.433i)15-s + (−0.988 + 0.149i)17-s + (0.5 + 0.866i)19-s + (0.988 + 0.149i)23-s + (0.826 − 0.563i)25-s + (−0.623 + 0.781i)27-s + (0.623 + 0.781i)29-s + (0.5 − 0.866i)31-s + (−0.733 + 0.680i)33-s + (0.365 − 0.930i)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: $\chi_{196} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 196,\ (1:\ ),\ 0.0320 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.868674333 + 1.809710173i\)
\(L(\frac12)\) \(\approx\) \(1.868674333 + 1.809710173i\)
\(L(1)\) \(\approx\) \(1.436661540 + 0.5670076693i\)
\(L(1)\) \(\approx\) \(1.436661540 + 0.5670076693i\)

Euler product

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

−26.48758156470736151976248762023, −25.45525711721110075231062177981, −24.65014327661099987686587549376, −24.07296240483551481255846565067, −22.61633785233617788803217486738, −21.69960031220947694489326920684, −20.78311829897036581804093990881, −19.70102160305605486039843681719, −18.90804384201568740389032650439, −17.854077271726110976362382283169, −17.21073231446941189756865477144, −15.64194975187990325934508164178, −14.6221131782625385896528381778, −13.645504098404661522517984989299, −13.15213673404593813908060840351, −11.81094247895112456828678180109, −10.520538772849798037539683598997, −9.35102132584066501768024474315, −8.51595339523109717847017816893, −7.17641716449742697756765443298, −6.33525438724190734712425105847, −4.99965250181824707086202329061, −3.111700544013634434968839662415, −2.3580045425302886450955766806, −0.813139909284623345149747165569, 1.75240806779827613440147869324, 2.75117753978303258728492852870, 4.38067984108830967020121507921, 5.173901449935731135864903597800, 6.7247303209863060624262848623, 7.99984415831730036811384385022, 9.31876023463218128667693525947, 9.73242167292922856422831257802, 10.86894303839565387963645220038, 12.43608116018026795438685458013, 13.40657249972901879712474137952, 14.402588819833973004106109847256, 15.144870279735067459968664513592, 16.37158353343906535960526484425, 17.23513406288797997512347156059, 18.28306800232461882992851326987, 19.62408009199064338503189854597, 20.35950375113051705440647427543, 21.27322579059675894725148703621, 21.971559723461335306268756224, 23.00699028269951831566869338453, 24.62071172442788690567881291447, 25.02443657134147877057501383476, 26.108765983547399699496224927536, 26.765494976054592706916165226834

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