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Properties

Label 1-163-163.25-r0-0-0
Degree $1$
Conductor $163$
Sign $-0.291 - 0.956i$
Analytic cond. $0.756968$
Root an. cond. $0.756968$
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  + (0.396 + 0.918i)2-s + (−0.286 − 0.957i)3-s + (−0.686 + 0.727i)4-s + (0.173 − 0.984i)5-s + (0.766 − 0.642i)6-s + (−0.993 − 0.116i)7-s + (−0.939 − 0.342i)8-s + (−0.835 + 0.549i)9-s + (0.973 − 0.230i)10-s + (−0.286 − 0.957i)11-s + (0.893 + 0.448i)12-s + (−0.939 + 0.342i)13-s + (−0.286 − 0.957i)14-s + (−0.993 + 0.116i)15-s + (−0.0581 − 0.998i)16-s + (−0.939 − 0.342i)17-s + ⋯
L(s)  = 1  + (0.396 + 0.918i)2-s + (−0.286 − 0.957i)3-s + (−0.686 + 0.727i)4-s + (0.173 − 0.984i)5-s + (0.766 − 0.642i)6-s + (−0.993 − 0.116i)7-s + (−0.939 − 0.342i)8-s + (−0.835 + 0.549i)9-s + (0.973 − 0.230i)10-s + (−0.286 − 0.957i)11-s + (0.893 + 0.448i)12-s + (−0.939 + 0.342i)13-s + (−0.286 − 0.957i)14-s + (−0.993 + 0.116i)15-s + (−0.0581 − 0.998i)16-s + (−0.939 − 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.291 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.291 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(163\)
Sign: $-0.291 - 0.956i$
Analytic conductor: \(0.756968\)
Root analytic conductor: \(0.756968\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{163} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 163,\ (0:\ ),\ -0.291 - 0.956i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3346871636 - 0.4517070096i\)
\(L(\frac12)\) \(\approx\) \(0.3346871636 - 0.4517070096i\)
\(L(1)\) \(\approx\) \(0.7552746251 - 0.1072683729i\)
\(L(1)\) \(\approx\) \(0.7552746251 - 0.1072683729i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad163 \( 1 \)
good2 \( 1 + (0.396 + 0.918i)T \)
3 \( 1 + (-0.286 - 0.957i)T \)
5 \( 1 + (0.173 - 0.984i)T \)
7 \( 1 + (-0.993 - 0.116i)T \)
11 \( 1 + (-0.286 - 0.957i)T \)
13 \( 1 + (-0.939 + 0.342i)T \)
17 \( 1 + (-0.939 - 0.342i)T \)
19 \( 1 + (0.597 + 0.802i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.396 - 0.918i)T \)
31 \( 1 + (0.173 - 0.984i)T \)
37 \( 1 + (0.766 + 0.642i)T \)
41 \( 1 + (-0.686 - 0.727i)T \)
43 \( 1 + (0.973 + 0.230i)T \)
47 \( 1 + (0.396 + 0.918i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + T \)
61 \( 1 + (-0.939 + 0.342i)T \)
67 \( 1 + (-0.686 - 0.727i)T \)
71 \( 1 + (0.973 + 0.230i)T \)
73 \( 1 + (-0.686 + 0.727i)T \)
79 \( 1 + (-0.835 - 0.549i)T \)
83 \( 1 + (-0.993 - 0.116i)T \)
89 \( 1 + (-0.286 + 0.957i)T \)
97 \( 1 + (-0.835 - 0.549i)T \)
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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

−28.30204744520505561307566376086, −27.064663451010300101187846079656, −26.41517530094898009140424917293, −25.4044390711221911710837940289, −23.601424800903817458886325818385, −22.75493065855666843563561141462, −22.03819732367560106294300818571, −21.61482287246207455040078451408, −20.032299549457664954812055743907, −19.67547217719232238691797756025, −18.14220663977657653088646860382, −17.46238727431050004347409944669, −15.70301688505038120870929643476, −15.095026096090335929365663287818, −14.0194990919671185797196500897, −12.77672785209267592115126232101, −11.71488700637000213550592934268, −10.59378659253127321409973471195, −9.9511417898961828423616927959, −9.15861308732596164115990537080, −6.98524982116274254255260398405, −5.73383395520082807304185181725, −4.58275405209800913248411829833, −3.318582619125758735615930303434, −2.46845476312088227266237442093, 0.40085838778439024724572964095, 2.64512450346352852010865954863, 4.36528055444774058084899361326, 5.67226675232651711119583064304, 6.3965408326331606173693709576, 7.62415426906728686548204806797, 8.581383927280241875332349940181, 9.7181504052017173874750584531, 11.74469562009222094029121980296, 12.61459604068712667555933084623, 13.392564950709352004293355983797, 14.11003134921490246992913419664, 15.80556697141970513968630586875, 16.58105474902724994491556376452, 17.23812427114443986689606733548, 18.451886265843972729192326435541, 19.40092211829390176055179168023, 20.63613350926056026927839185508, 22.07243603233818448829252093717, 22.7000093857345344688279380300, 23.98375284966092793590114484116, 24.3666953485817137542039726643, 25.17874215076852373540900832672, 26.23482973550453685706374088376, 27.23744460068233040759795069422

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