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Properties

Label 1-163-163.38-r0-0-0
Degree $1$
Conductor $163$
Sign $-0.367 + 0.929i$
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  + (0.173 − 0.984i)2-s + (−0.939 − 0.342i)3-s + (−0.939 − 0.342i)4-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)6-s + (0.173 − 0.984i)7-s + (−0.5 + 0.866i)8-s + (0.766 + 0.642i)9-s + (−0.939 + 0.342i)10-s + (−0.939 − 0.342i)11-s + (0.766 + 0.642i)12-s + (−0.5 − 0.866i)13-s + (−0.939 − 0.342i)14-s + (0.173 + 0.984i)15-s + (0.766 + 0.642i)16-s + (−0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)2-s + (−0.939 − 0.342i)3-s + (−0.939 − 0.342i)4-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)6-s + (0.173 − 0.984i)7-s + (−0.5 + 0.866i)8-s + (0.766 + 0.642i)9-s + (−0.939 + 0.342i)10-s + (−0.939 − 0.342i)11-s + (0.766 + 0.642i)12-s + (−0.5 − 0.866i)13-s + (−0.939 − 0.342i)14-s + (0.173 + 0.984i)15-s + (0.766 + 0.642i)16-s + (−0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.367 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.367 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(163\)
Sign: $-0.367 + 0.929i$
Analytic conductor: \(0.756968\)
Root analytic conductor: \(0.756968\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{163} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 163,\ (0:\ ),\ -0.367 + 0.929i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2013473672 - 0.2960863448i\)
\(L(\frac12)\) \(\approx\) \(-0.2013473672 - 0.2960863448i\)
\(L(1)\) \(\approx\) \(0.3215872627 - 0.4724477912i\)
\(L(1)\) \(\approx\) \(0.3215872627 - 0.4724477912i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad163 \( 1 \)
good2 \( 1 + (0.173 - 0.984i)T \)
3 \( 1 + (-0.939 - 0.342i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (0.173 - 0.984i)T \)
11 \( 1 + (-0.939 - 0.342i)T \)
13 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (0.173 + 0.984i)T \)
23 \( 1 + T \)
29 \( 1 + (0.173 + 0.984i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (-0.939 + 0.342i)T \)
43 \( 1 + (-0.939 - 0.342i)T \)
47 \( 1 + (0.173 - 0.984i)T \)
53 \( 1 + T \)
59 \( 1 + T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (-0.939 + 0.342i)T \)
71 \( 1 + (-0.939 - 0.342i)T \)
73 \( 1 + (-0.939 - 0.342i)T \)
79 \( 1 + (0.766 - 0.642i)T \)
83 \( 1 + (0.173 - 0.984i)T \)
89 \( 1 + (-0.939 + 0.342i)T \)
97 \( 1 + (0.766 - 0.642i)T \)
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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

−28.27185017760771753380810602038, −27.136393895041585865875731706210, −26.6240306829842338509766573318, −25.54332228801351240607314145965, −24.30166156714602018581897325667, −23.5607807851848705304634616635, −22.63929142578391573876790076281, −21.9748753368191618559554834503, −21.120601076741124553738536598824, −19.07196218370057615571325627635, −18.26469083162473068221705785208, −17.58998218736321011933682920619, −16.2813008475645464543995394326, −15.4467548905179036586065888657, −14.955542758595490353280215088, −13.48825705986945550590844091341, −12.16591937763637657381804286373, −11.35202420772798185833739116266, −9.976746804791124771090400880865, −8.83920523671517001814915632653, −7.2948928622281133877209219868, −6.63561331583509030678592406068, −5.28458431965108260629223073065, −4.546616462037911815401691060302, −2.84886005171284635133767906502, 0.323316169180063361464340658411, 1.61904673805004152161338785720, 3.60159157112091480912037118619, 4.79986617085908685106775235396, 5.57682102139356032623958326608, 7.44903748316190742167010151129, 8.47756609098513786266577776011, 10.20010208730030685972737910749, 10.78734575254920642848907895637, 11.89697391840432230018148350714, 12.90332875848988156198025143676, 13.33433335218173070087625252082, 15.02261893476488562423304264751, 16.474316546052276885365211166548, 17.25283639614134065124615235478, 18.25246326154287468443845645943, 19.34902723934731624305247549769, 20.25333554557570374672509632680, 21.105197465547585737544167971576, 22.23722568485955240264341687988, 23.345134900220570739603794809585, 23.664717234422748543121161077858, 24.72936522662241496571771572900, 26.65491985712585502012051166098, 27.34556775880318423982689514390

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