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Properties

Label 1-403-403.60-r0-0-0
Degree $1$
Conductor $403$
Sign $-0.311 - 0.950i$
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  + (−0.951 + 0.309i)2-s + (−0.309 + 0.951i)3-s + (0.809 − 0.587i)4-s + i·5-s − i·6-s + (0.587 + 0.809i)7-s + (−0.587 + 0.809i)8-s + (−0.809 − 0.587i)9-s + (−0.309 − 0.951i)10-s + (−0.587 − 0.809i)11-s + (0.309 + 0.951i)12-s + (−0.809 − 0.587i)14-s + (−0.951 − 0.309i)15-s + (0.309 − 0.951i)16-s + (−0.809 − 0.587i)17-s + (0.951 + 0.309i)18-s + ⋯
L(s)  = 1  + (−0.951 + 0.309i)2-s + (−0.309 + 0.951i)3-s + (0.809 − 0.587i)4-s + i·5-s − i·6-s + (0.587 + 0.809i)7-s + (−0.587 + 0.809i)8-s + (−0.809 − 0.587i)9-s + (−0.309 − 0.951i)10-s + (−0.587 − 0.809i)11-s + (0.309 + 0.951i)12-s + (−0.809 − 0.587i)14-s + (−0.951 − 0.309i)15-s + (0.309 − 0.951i)16-s + (−0.809 − 0.587i)17-s + (0.951 + 0.309i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.311 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.311 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.311 - 0.950i$
Analytic conductor: \(1.87152\)
Root analytic conductor: \(1.87152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (60, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 403,\ (0:\ ),\ -0.311 - 0.950i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.08506073533 + 0.1174162801i\)
\(L(\frac12)\) \(\approx\) \(-0.08506073533 + 0.1174162801i\)
\(L(1)\) \(\approx\) \(0.3864520842 + 0.2846898548i\)
\(L(1)\) \(\approx\) \(0.3864520842 + 0.2846898548i\)

Euler product

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

−23.97395010278739140255621915174, −23.250673697666925507705750689204, −21.78636959598519659147315781591, −20.817233158028061662726062142849, −19.97627934164879936687003256493, −19.612274884556599958185219349739, −18.30811347171966406801167861482, −17.58807766522550348786463748409, −17.13340725616142600977016723684, −16.234578191443680474964542688466, −15.09786321846852647324790629409, −13.612620337771200707752595374071, −12.82331565319121700732833143593, −12.156408775174121707937668849303, −11.09577023879093470799743996066, −10.363112815374695802271016249133, −9.03767847263623019925897062236, −8.18313404282211836871231697261, −7.49656973696156243856382385189, −6.56312805011822689700490897708, −5.21622458792146458755960866456, −3.99455239622960168448337425441, −2.13146425905859053354390416380, −1.53100202649399403734264552064, −0.11318437959023882263430688942, 2.1756002884574737489100431322, 3.06371168976523119067230580377, 4.66404420556087179222261924283, 5.87982138351855146821548325958, 6.43014105923293396207553658123, 7.93826502119954790755972730395, 8.64327083318023655116904909764, 9.70524297552319932334829890657, 10.568604487636408381550607579942, 11.21737465420692280406257111786, 11.906261824762211069264656152963, 13.83295545158578312412356011014, 14.87993804281706107486907425887, 15.32022161339953091219162168818, 16.16240291419854004199955797504, 17.1065525949902651633000041677, 18.109611917630123294599770822038, 18.53634707366774518868552785370, 19.58909614893202858174109219165, 20.732518513611314271073974993422, 21.4315135177212334910981275124, 22.240700059969352818622544086372, 23.25659071725007513514397319396, 24.221334559751961173408099688485, 25.16803002154146654291677328448

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