Properties

Label 1-203-203.11-r1-0-0
Degree $1$
Conductor $203$
Sign $-0.892 + 0.451i$
Analytic cond. $21.8153$
Root an. cond. $21.8153$
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  + (0.149 + 0.988i)2-s + (0.680 + 0.733i)3-s + (−0.955 + 0.294i)4-s + (0.988 − 0.149i)5-s + (−0.623 + 0.781i)6-s + (−0.433 − 0.900i)8-s + (−0.0747 + 0.997i)9-s + (0.294 + 0.955i)10-s + (0.997 − 0.0747i)11-s + (−0.866 − 0.5i)12-s + (0.900 + 0.433i)13-s + (0.781 + 0.623i)15-s + (0.826 − 0.563i)16-s + (−0.866 + 0.5i)17-s + (−0.997 + 0.0747i)18-s + (−0.680 + 0.733i)19-s + ⋯
L(s)  = 1  + (0.149 + 0.988i)2-s + (0.680 + 0.733i)3-s + (−0.955 + 0.294i)4-s + (0.988 − 0.149i)5-s + (−0.623 + 0.781i)6-s + (−0.433 − 0.900i)8-s + (−0.0747 + 0.997i)9-s + (0.294 + 0.955i)10-s + (0.997 − 0.0747i)11-s + (−0.866 − 0.5i)12-s + (0.900 + 0.433i)13-s + (0.781 + 0.623i)15-s + (0.826 − 0.563i)16-s + (−0.866 + 0.5i)17-s + (−0.997 + 0.0747i)18-s + (−0.680 + 0.733i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.892 + 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.892 + 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(203\)    =    \(7 \cdot 29\)
Sign: $-0.892 + 0.451i$
Analytic conductor: \(21.8153\)
Root analytic conductor: \(21.8153\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{203} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 203,\ (1:\ ),\ -0.892 + 0.451i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6450913809 + 2.700882777i\)
\(L(\frac12)\) \(\approx\) \(0.6450913809 + 2.700882777i\)
\(L(1)\) \(\approx\) \(1.069937682 + 1.203457904i\)
\(L(1)\) \(\approx\) \(1.069937682 + 1.203457904i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.149 + 0.988i)T \)
3 \( 1 + (0.680 + 0.733i)T \)
5 \( 1 + (0.988 - 0.149i)T \)
11 \( 1 + (0.997 - 0.0747i)T \)
13 \( 1 + (0.900 + 0.433i)T \)
17 \( 1 + (-0.866 + 0.5i)T \)
19 \( 1 + (-0.680 + 0.733i)T \)
23 \( 1 + (0.365 + 0.930i)T \)
31 \( 1 + (-0.930 - 0.365i)T \)
37 \( 1 + (0.997 + 0.0747i)T \)
41 \( 1 + iT \)
43 \( 1 + (-0.781 - 0.623i)T \)
47 \( 1 + (0.563 + 0.826i)T \)
53 \( 1 + (0.365 - 0.930i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (0.294 - 0.955i)T \)
67 \( 1 + (-0.826 - 0.563i)T \)
71 \( 1 + (0.900 + 0.433i)T \)
73 \( 1 + (0.149 - 0.988i)T \)
79 \( 1 + (-0.997 - 0.0747i)T \)
83 \( 1 + (-0.222 + 0.974i)T \)
89 \( 1 + (0.149 + 0.988i)T \)
97 \( 1 + (-0.974 - 0.222i)T \)
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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

−26.16378582130755519929155421149, −25.27825394676612254325227102618, −24.40721114108351929174284626894, −23.23191927135720293266683842501, −22.258073431474202135053971066084, −21.34077264938531174053643425897, −20.36063931640999981457814318363, −19.75508452458418880047042019618, −18.54506747652545181137901853455, −18.005594574179009870409185013347, −17.03641494725316422147471293874, −15.06236803762134702603467208076, −14.1999646734083766013058387668, −13.366159150271532780613500245901, −12.757510265288386565015943995, −11.484853625398333460959619832305, −10.42975817995473871927551773862, −9.09021353971210151187933509831, −8.74515367270167758343205745344, −6.90149790210482947473222719410, −5.885726457873775223082858418185, −4.276100927509120323391304741616, −2.95203139866434991511851923358, −2.001301581147698971778056728715, −0.88260303353937249420991484008, 1.71950773953182297509789730676, 3.54376386640105995502311239483, 4.46966128203417944510124565892, 5.7592650651146416683670156231, 6.652059921066350551474233225158, 8.20650638254814431805081334514, 9.05869752889049114864640780989, 9.69848222696895073334799226500, 11.05455763294338529382328971730, 12.90448338202204535423382729023, 13.69105044830659964093240342749, 14.48742792100552244092111236450, 15.30480776150218581808649588244, 16.495497301870292191961340531676, 17.04521728386443927870737171991, 18.211048659347984324540930385161, 19.341663796400842737120711652542, 20.5991100541234060889804209967, 21.638423266025996964868831592004, 22.04708739848037737730353006593, 23.31383194471887759735651255724, 24.4985464420930898253028739664, 25.36908890955111770492200464805, 25.75072658875368438118918443738, 26.81128772628936095179296189076

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