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Properties

Label 1-207-207.164-r1-0-0
Degree $1$
Conductor $207$
Sign $0.935 + 0.352i$
Analytic cond. $22.2452$
Root an. cond. $22.2452$
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  + (−0.723 − 0.690i)2-s + (0.0475 + 0.998i)4-s + (−0.928 − 0.371i)5-s + (−0.995 + 0.0950i)7-s + (0.654 − 0.755i)8-s + (0.415 + 0.909i)10-s + (−0.235 − 0.971i)11-s + (−0.995 − 0.0950i)13-s + (0.786 + 0.618i)14-s + (−0.995 + 0.0950i)16-s + (−0.841 − 0.540i)17-s + (0.841 − 0.540i)19-s + (0.327 − 0.945i)20-s + (−0.5 + 0.866i)22-s + (0.723 + 0.690i)25-s + (0.654 + 0.755i)26-s + ⋯
L(s)  = 1  + (−0.723 − 0.690i)2-s + (0.0475 + 0.998i)4-s + (−0.928 − 0.371i)5-s + (−0.995 + 0.0950i)7-s + (0.654 − 0.755i)8-s + (0.415 + 0.909i)10-s + (−0.235 − 0.971i)11-s + (−0.995 − 0.0950i)13-s + (0.786 + 0.618i)14-s + (−0.995 + 0.0950i)16-s + (−0.841 − 0.540i)17-s + (0.841 − 0.540i)19-s + (0.327 − 0.945i)20-s + (−0.5 + 0.866i)22-s + (0.723 + 0.690i)25-s + (0.654 + 0.755i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.935 + 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.935 + 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(207\)    =    \(3^{2} \cdot 23\)
Sign: $0.935 + 0.352i$
Analytic conductor: \(22.2452\)
Root analytic conductor: \(22.2452\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{207} (164, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 207,\ (1:\ ),\ 0.935 + 0.352i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3578240391 + 0.06509262351i\)
\(L(\frac12)\) \(\approx\) \(0.3578240391 + 0.06509262351i\)
\(L(1)\) \(\approx\) \(0.4502666978 - 0.1621178959i\)
\(L(1)\) \(\approx\) \(0.4502666978 - 0.1621178959i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.723 - 0.690i)T \)
5 \( 1 + (-0.928 - 0.371i)T \)
7 \( 1 + (-0.995 + 0.0950i)T \)
11 \( 1 + (-0.235 - 0.971i)T \)
13 \( 1 + (-0.995 - 0.0950i)T \)
17 \( 1 + (-0.841 - 0.540i)T \)
19 \( 1 + (0.841 - 0.540i)T \)
29 \( 1 + (-0.0475 + 0.998i)T \)
31 \( 1 + (-0.327 - 0.945i)T \)
37 \( 1 + (-0.142 + 0.989i)T \)
41 \( 1 + (-0.928 - 0.371i)T \)
43 \( 1 + (-0.327 + 0.945i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (-0.415 + 0.909i)T \)
59 \( 1 + (0.995 + 0.0950i)T \)
61 \( 1 + (0.981 + 0.189i)T \)
67 \( 1 + (0.235 - 0.971i)T \)
71 \( 1 + (0.959 - 0.281i)T \)
73 \( 1 + (0.841 - 0.540i)T \)
79 \( 1 + (0.580 - 0.814i)T \)
83 \( 1 + (-0.928 + 0.371i)T \)
89 \( 1 + (0.654 + 0.755i)T \)
97 \( 1 + (-0.786 + 0.618i)T \)
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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

−26.651628734471224609590928234736, −25.657988275058160667550896247631, −24.741687350319751015060070482199, −23.68534049404425528439083059178, −22.90684233342091838048184694069, −22.16884778603422614565495158601, −20.2632208113838836879248388229, −19.66756138698201477101919527491, −18.88142438499773123577763170112, −17.86014681580771547977761951644, −16.83972697147466742937811408284, −15.82212436752363287048862193771, −15.23805954948780847448501905987, −14.22981577760451227404941917184, −12.79199864677202505550890191251, −11.68975622593573038202928247935, −10.36006724358196531996828581312, −9.68447987563231837126630010301, −8.408030583028890555555026370632, −7.26096657320853395940944569601, −6.75659626744310866394206333555, −5.23931789317168769657062482642, −3.896092749906599690561476951600, −2.25912241268507505251056937614, −0.24935551342412438739888039740, 0.71189851887497253144600070075, 2.693300506665728327567645694270, 3.55197669179213775546922625660, 4.93617872336680385527272029611, 6.77899046525937679215606725850, 7.74796927151584429421037677365, 8.86341968704299412402475801359, 9.65425254055098474298128916488, 10.941293422125955155453011205664, 11.78555564073177963459057960235, 12.70593937278005955311047156801, 13.60306076056984558100461952123, 15.41379251541612317324220049418, 16.202466902476378137320829193150, 16.933304939784128792094486385090, 18.28194869989912764054231957835, 19.13824218422759450102566726886, 19.83904156641043769956240932396, 20.53203361220426300955241755644, 22.01591838871216312985521984400, 22.40052911059463161442655704939, 23.88629853468637178839344086373, 24.75676575693412430961683415895, 25.984305010161482610765354211448, 26.814517675070984611513098186144

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