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Properties

Label 1-403-403.158-r0-0-0
Degree $1$
Conductor $403$
Sign $-0.997 - 0.0759i$
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 + (−0.669 + 0.743i)3-s + (0.104 − 0.994i)4-s + (−0.866 + 0.5i)5-s + i·6-s + (0.587 − 0.809i)7-s + (−0.587 − 0.809i)8-s + (−0.104 − 0.994i)9-s + (−0.309 + 0.951i)10-s + (−0.587 + 0.809i)11-s + (0.669 + 0.743i)12-s + (−0.104 − 0.994i)14-s + (0.207 − 0.978i)15-s + (−0.978 − 0.207i)16-s + (−0.809 + 0.587i)17-s + (−0.743 − 0.669i)18-s + ⋯
L(s)  = 1  + (0.743 − 0.669i)2-s + (−0.669 + 0.743i)3-s + (0.104 − 0.994i)4-s + (−0.866 + 0.5i)5-s + i·6-s + (0.587 − 0.809i)7-s + (−0.587 − 0.809i)8-s + (−0.104 − 0.994i)9-s + (−0.309 + 0.951i)10-s + (−0.587 + 0.809i)11-s + (0.669 + 0.743i)12-s + (−0.104 − 0.994i)14-s + (0.207 − 0.978i)15-s + (−0.978 − 0.207i)16-s + (−0.809 + 0.587i)17-s + (−0.743 − 0.669i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0759i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0759i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.997 - 0.0759i$
Analytic conductor: \(1.87152\)
Root analytic conductor: \(1.87152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (158, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 403,\ (0:\ ),\ -0.997 - 0.0759i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01565774325 - 0.4118781554i\)
\(L(\frac12)\) \(\approx\) \(0.01565774325 - 0.4118781554i\)
\(L(1)\) \(\approx\) \(0.7660244189 - 0.2753696220i\)
\(L(1)\) \(\approx\) \(0.7660244189 - 0.2753696220i\)

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 \( 1 + (-0.669 + 0.743i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
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 \( 1 + (-0.104 - 0.994i)T \)
29 \( 1 + (-0.669 - 0.743i)T \)
37 \( 1 + (-0.866 + 0.5i)T \)
41 \( 1 + (-0.951 - 0.309i)T \)
43 \( 1 + (0.309 - 0.951i)T \)
47 \( 1 + (-0.951 + 0.309i)T \)
53 \( 1 + (-0.913 - 0.406i)T \)
59 \( 1 + (-0.951 + 0.309i)T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 - iT \)
71 \( 1 + (-0.994 + 0.104i)T \)
73 \( 1 + (0.994 + 0.104i)T \)
79 \( 1 + (0.104 + 0.994i)T \)
83 \( 1 + (-0.207 - 0.978i)T \)
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

−24.58280338061777985940007752254, −23.81066793382724850008877367048, −23.40730946439131236339606607473, −22.345968694977880200572402016391, −21.587551730811032179634492419070, −20.66729422202190773684833902430, −19.44945874357527268517991397642, −18.50840719812790517152025378056, −17.69388161453612054678501161589, −16.72580785377971119936449805517, −15.94810185719756674641105898694, −15.25756148687824552367243724586, −14.096521299665695733054906753243, −13.07793973276742368538372990636, −12.4829949600386320582482366312, −11.4952432781110544224388346066, −11.09756392731954493050776094294, −8.85328145220458587318757149529, −8.15481899982603645558195327152, −7.387239040419232731568135163751, −6.2599244702776941112781303921, −5.31795900596875230371786963210, −4.68658424323422594232918443, −3.25531305112101190494485609488, −1.889179903632552690822213674062, 0.192312678931898196495004033951, 2.04222012060277424141817906948, 3.47472218011610766702796524063, 4.38059552308153146699844481848, 4.80566747592293066741064935519, 6.28161713729161195291288083875, 7.136108574178575301503582729195, 8.571925545890367240011431938551, 10.081263650983930436676460025605, 10.64410194702510390926560646799, 11.24928375373295248323555286831, 12.16609698221734205301218127934, 13.07339480129253432134583470944, 14.35623293720229772805133674762, 15.15577252807306858466915217991, 15.59846117836434825800757893597, 16.898516334209960038674982014045, 17.84757640391715924026279893734, 18.837067669566061992674286887602, 19.93560740940151808316969757011, 20.582150742001232953131926144887, 21.314593884716328048186402779263, 22.45519897153593039795699792788, 22.8008179210170432961556025744, 23.80918124963483738695560513695

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