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Properties

Label 1-403-403.24-r0-0-0
Degree $1$
Conductor $403$
Sign $0.0168 - 0.999i$
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  + (0.994 − 0.104i)2-s + (0.104 − 0.994i)3-s + (0.978 − 0.207i)4-s + (0.866 − 0.5i)5-s − i·6-s + (−0.951 − 0.309i)7-s + (0.951 − 0.309i)8-s + (−0.978 − 0.207i)9-s + (0.809 − 0.587i)10-s + (0.951 + 0.309i)11-s + (−0.104 − 0.994i)12-s + (−0.978 − 0.207i)14-s + (−0.406 − 0.913i)15-s + (0.913 − 0.406i)16-s + (0.309 + 0.951i)17-s + (−0.994 − 0.104i)18-s + ⋯
L(s)  = 1  + (0.994 − 0.104i)2-s + (0.104 − 0.994i)3-s + (0.978 − 0.207i)4-s + (0.866 − 0.5i)5-s − i·6-s + (−0.951 − 0.309i)7-s + (0.951 − 0.309i)8-s + (−0.978 − 0.207i)9-s + (0.809 − 0.587i)10-s + (0.951 + 0.309i)11-s + (−0.104 − 0.994i)12-s + (−0.978 − 0.207i)14-s + (−0.406 − 0.913i)15-s + (0.913 − 0.406i)16-s + (0.309 + 0.951i)17-s + (−0.994 − 0.104i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0168 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0168 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.0168 - 0.999i$
Analytic conductor: \(1.87152\)
Root analytic conductor: \(1.87152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 403,\ (0:\ ),\ 0.0168 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.900018234 - 1.932356242i\)
\(L(\frac12)\) \(\approx\) \(1.900018234 - 1.932356242i\)
\(L(1)\) \(\approx\) \(1.802374646 - 0.9932658418i\)
\(L(1)\) \(\approx\) \(1.802374646 - 0.9932658418i\)

Euler product

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

−24.959112496727890913890610990741, −23.31112691476472667977933865893, −22.59818274390737813545693257466, −21.9907401722166619305272712408, −21.460625186447642199470148761094, −20.49984433475418359375275278102, −19.673642317674598512312009965149, −18.60216498085848335394835451651, −17.09291367468182602184891574449, −16.54341590257294406869867367608, −15.65086055814226770404749506301, −14.74503983896412489224444704945, −14.06580945748266543298862755765, −13.313814807443051902176303670, −12.044455471321727622572788816167, −11.2586178095376685056233964881, −10.02229858015598006458371727722, −9.62271301316481587376373741147, −8.20864902132962353613643684065, −6.5939693833188473113916498494, −6.085108265552407585256000659553, −5.11892122054008689159282287279, −3.8425618112530182090273371438, −3.13999331918138751543876788026, −2.084539406190434790762380182009, 1.24421728039599331177419826786, 2.17362599360826347807129755048, 3.32960114117137747644856686226, 4.52374337027819740422552374857, 5.948426278068327630511178080545, 6.35533446229563440168536808263, 7.27103899789314154459459399391, 8.65253217326789049070521668182, 9.75048238018908095599767711809, 10.831523896423945899290017805434, 12.14421423299977785209254095146, 12.67033219201404396332541304622, 13.373737180075465880164664882185, 14.1281339788618399717773352281, 14.95301134443155422067366586090, 16.38175854384208809698902581576, 16.98798125420450784709026546978, 17.952507115512223769047003861295, 19.3438101773580952279840543254, 19.78181967318441593109428681331, 20.59503922913353722745047793070, 21.825549235586392832405847204507, 22.325582364824076829419955353677, 23.44868317030470442267623524918, 23.99286886542146750695674641063

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