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Properties

Label 1-14e2-196.95-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: $\chi_{196} (95, \cdot )$
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

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

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