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Properties

Label 1-403-403.15-r0-0-0
Degree $1$
Conductor $403$
Sign $-0.502 - 0.864i$
Analytic cond. $1.87152$
Root an. cond. $1.87152$
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  + (0.743 − 0.669i)2-s + (0.978 + 0.207i)3-s + (0.104 − 0.994i)4-s − i·5-s + (0.866 − 0.5i)6-s + (−0.994 − 0.104i)7-s + (−0.587 − 0.809i)8-s + (0.913 + 0.406i)9-s + (−0.669 − 0.743i)10-s + (−0.406 − 0.913i)11-s + (0.309 − 0.951i)12-s + (−0.809 + 0.587i)14-s + (0.207 − 0.978i)15-s + (−0.978 − 0.207i)16-s + (0.913 + 0.406i)17-s + (0.951 − 0.309i)18-s + ⋯
L(s)  = 1  + (0.743 − 0.669i)2-s + (0.978 + 0.207i)3-s + (0.104 − 0.994i)4-s − i·5-s + (0.866 − 0.5i)6-s + (−0.994 − 0.104i)7-s + (−0.587 − 0.809i)8-s + (0.913 + 0.406i)9-s + (−0.669 − 0.743i)10-s + (−0.406 − 0.913i)11-s + (0.309 − 0.951i)12-s + (−0.809 + 0.587i)14-s + (0.207 − 0.978i)15-s + (−0.978 − 0.207i)16-s + (0.913 + 0.406i)17-s + (0.951 − 0.309i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.502 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.502 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.502 - 0.864i$
Analytic conductor: \(1.87152\)
Root analytic conductor: \(1.87152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 403,\ (0:\ ),\ -0.502 - 0.864i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.168715041 - 2.031007433i\)
\(L(\frac12)\) \(\approx\) \(1.168715041 - 2.031007433i\)
\(L(1)\) \(\approx\) \(1.476245947 - 1.098724622i\)
\(L(1)\) \(\approx\) \(1.476245947 - 1.098724622i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.743 - 0.669i)T \)
3 \( 1 + (0.978 + 0.207i)T \)
5 \( 1 - iT \)
7 \( 1 + (-0.994 - 0.104i)T \)
11 \( 1 + (-0.406 - 0.913i)T \)
17 \( 1 + (0.913 + 0.406i)T \)
19 \( 1 + (0.207 + 0.978i)T \)
23 \( 1 + (-0.104 - 0.994i)T \)
29 \( 1 + (-0.669 - 0.743i)T \)
37 \( 1 + (0.866 + 0.5i)T \)
41 \( 1 + (0.743 - 0.669i)T \)
43 \( 1 + (-0.978 + 0.207i)T \)
47 \( 1 + (-0.951 + 0.309i)T \)
53 \( 1 + (0.809 - 0.587i)T \)
59 \( 1 + (0.743 + 0.669i)T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 + (0.406 - 0.913i)T \)
73 \( 1 + (-0.587 + 0.809i)T \)
79 \( 1 + (0.809 - 0.587i)T \)
83 \( 1 + (0.951 + 0.309i)T \)
89 \( 1 + (-0.406 - 0.913i)T \)
97 \( 1 + (0.994 + 0.104i)T \)
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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

−24.91036595870936121261910992265, −23.56945703998751163254753119534, −23.11429476639503162901086398127, −22.06576462149208184446637489018, −21.445075369729640159166773189401, −20.30915771206203519418848527241, −19.54162105883585313648222964472, −18.45303078842457306891745985449, −17.80770438389660918450599897541, −16.396783940784277884435532992890, −15.500596734277834566822168862722, −14.9644429717575264321755888122, −14.10852166910231316488898580021, −13.2585288077938619556434795122, −12.60346830703310634790820724730, −11.44684197857853733698019026017, −9.960528557416509921399314161, −9.2682229917034251772765878373, −7.83137931819793056410072594426, −7.20829024560726369438630622810, −6.508498128438164383792363128700, −5.22080382671571424971888067313, −3.79082001004596026558799949611, −3.09553336542216971085613031338, −2.25934527701480507106904011885, 0.98486239084903842484967170494, 2.33491193304830983471212287043, 3.42870928675167639723382040053, 4.07333544856265139148508652875, 5.334844233014065015520264452388, 6.25588344645866927940497890091, 7.82815221906630785094666285815, 8.78765564118330912730840107767, 9.78326032879503181776173408177, 10.331351526049743079950017782325, 11.79047603798244743974962719962, 12.8202333400634524588469547340, 13.212355705910448779373973357697, 14.13625115862233869572698718930, 15.04251030197278647211013885042, 16.176246885994258444370416223936, 16.51894029305987779858213746795, 18.51472109594001901410362266, 19.16284346012725114045251680366, 19.85847973121414073278124849874, 20.768138376030757119748549324137, 21.16378583089985257108808253514, 22.17029922876262523479379959643, 23.14908405191290496755103312747, 24.11994910542545644901743038073

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