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Properties

Label 1-14e2-196.163-r1-0-0
Degree $1$
Conductor $196$
Sign $-0.0534 - 0.998i$
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  + (0.988 − 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

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

Invariants

Degree: \(1\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $-0.0534 - 0.998i$
Analytic conductor: \(21.0631\)
Root analytic conductor: \(21.0631\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{196} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 196,\ (1:\ ),\ -0.0534 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.753810978 - 1.850109832i\)
\(L(\frac12)\) \(\approx\) \(1.753810978 - 1.850109832i\)
\(L(1)\) \(\approx\) \(1.428580744 - 0.5594265586i\)
\(L(1)\) \(\approx\) \(1.428580744 - 0.5594265586i\)

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

−26.51562436998477179180328051600, −26.17823438646677564922657718753, −25.40930398038790359572101853392, −24.204300730904378298283290141156, −23.30025459265424964423533707835, −21.892231566652723034792421191846, −21.41260111877027836448147397166, −20.32894303692624157445476165711, −19.23988359261874429865958860222, −18.5637273530435801841720648260, −17.55728344881939819630112094067, −16.07779270923688589723564203165, −15.21420827545114345260916159968, −14.248319908027459185624301091768, −13.623894298866144749039491839853, −12.38341213705132891491799778589, −10.90240290401676601405927843168, −9.99897376582789524813432728046, −9.11224302723832107838932383838, −7.72261501911041248154015938222, −7.00738521154712341609741705671, −5.47992550320566125526956615875, −3.97889354668952464446275639831, −2.83714359656793396311095773702, −1.85019175322524561782177681621, 0.7677830582652272451211069975, 2.216029300907720672303039495224, 3.4149990992093451788501583337, 4.85452046879823247359900411334, 5.95076361564232498640501710335, 7.80819639000542141019535748203, 8.1473251891001104131552534640, 9.5635220865424307908972069573, 10.18544841687678269643423385455, 11.99625447149901389591910466856, 12.96860268160851602075776522194, 13.63622321369080240161864630288, 14.7468618834188833351163174808, 15.82595797402970800182498768018, 16.68554481889957111168695459198, 18.06503188648960199672387455141, 18.79532890203141618104495656355, 20.13160233085545899039575106568, 20.61246974070751967773873998451, 21.39644287643011394137283924876, 22.746050102417439380757971351227, 24.00217654292068257721844154422, 24.69685522450958514387269945209, 25.44384687313554097220150501350, 26.38687248857577204456078863900

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