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Properties

Label 1-451-451.294-r0-0-0
Degree $1$
Conductor $451$
Sign $0.528 - 0.848i$
Analytic cond. $2.09443$
Root an. cond. $2.09443$
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  + (−0.587 − 0.809i)2-s + (0.987 + 0.156i)3-s + (−0.309 + 0.951i)4-s + (−0.587 − 0.809i)5-s + (−0.453 − 0.891i)6-s + (0.707 + 0.707i)7-s + (0.951 − 0.309i)8-s + (0.951 + 0.309i)9-s + (−0.309 + 0.951i)10-s + (−0.453 + 0.891i)12-s + (−0.987 − 0.156i)13-s + (0.156 − 0.987i)14-s + (−0.453 − 0.891i)15-s + (−0.809 − 0.587i)16-s + (0.707 − 0.707i)17-s + (−0.309 − 0.951i)18-s + ⋯
L(s)  = 1  + (−0.587 − 0.809i)2-s + (0.987 + 0.156i)3-s + (−0.309 + 0.951i)4-s + (−0.587 − 0.809i)5-s + (−0.453 − 0.891i)6-s + (0.707 + 0.707i)7-s + (0.951 − 0.309i)8-s + (0.951 + 0.309i)9-s + (−0.309 + 0.951i)10-s + (−0.453 + 0.891i)12-s + (−0.987 − 0.156i)13-s + (0.156 − 0.987i)14-s + (−0.453 − 0.891i)15-s + (−0.809 − 0.587i)16-s + (0.707 − 0.707i)17-s + (−0.309 − 0.951i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 451 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.528 - 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 451 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.528 - 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(451\)    =    \(11 \cdot 41\)
Sign: $0.528 - 0.848i$
Analytic conductor: \(2.09443\)
Root analytic conductor: \(2.09443\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{451} (294, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 451,\ (0:\ ),\ 0.528 - 0.848i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.179531277 - 0.6547023027i\)
\(L(\frac12)\) \(\approx\) \(1.179531277 - 0.6547023027i\)
\(L(1)\) \(\approx\) \(1.018300727 - 0.3760081425i\)
\(L(1)\) \(\approx\) \(1.018300727 - 0.3760081425i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.587 - 0.809i)T \)
3 \( 1 + (0.987 + 0.156i)T \)
5 \( 1 + (-0.587 - 0.809i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
13 \( 1 + (-0.987 - 0.156i)T \)
17 \( 1 + (0.707 - 0.707i)T \)
19 \( 1 + (-0.453 - 0.891i)T \)
23 \( 1 + (0.809 + 0.587i)T \)
29 \( 1 + (0.891 - 0.453i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + (0.309 - 0.951i)T \)
43 \( 1 + (0.587 - 0.809i)T \)
47 \( 1 + (-0.707 + 0.707i)T \)
53 \( 1 + (0.707 + 0.707i)T \)
59 \( 1 + (0.309 + 0.951i)T \)
61 \( 1 + (0.587 - 0.809i)T \)
67 \( 1 + (-0.891 - 0.453i)T \)
71 \( 1 + (-0.707 - 0.707i)T \)
73 \( 1 + (0.587 - 0.809i)T \)
79 \( 1 + (-0.891 + 0.453i)T \)
83 \( 1 + (-0.309 - 0.951i)T \)
89 \( 1 + (0.987 + 0.156i)T \)
97 \( 1 + (0.707 - 0.707i)T \)
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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

−24.227299551197409892611167443880, −23.54315711900410114371962175682, −22.75251610784747880451796343093, −21.458022174180415116578686398075, −20.45749209295420344857058343415, −19.48463386713863571255407362181, −19.066742320629500486097487651439, −18.19409200833958109154440653544, −17.230288526775810239180850117239, −16.34412434438372336445599725571, −15.15935868733916522943867555761, −14.59449000240561247271422628399, −14.23793284153103685098708956835, −13.00130302693100929756795467021, −11.66502049628040984650271818692, −10.34870386594294640665631817443, −10.01300671908962577417047296890, −8.50800478299416426488871311765, −7.96763222492385223056682543057, −7.21610481132858141950085672816, −6.44015565038976563936987004748, −4.78895372834678414543674576993, −3.8992349898770444730010791312, −2.519782926900288095344652341, −1.21595067856673900456290629439, 1.04386508048470418534521561395, 2.30037115523701458566017918236, 3.10744183059619717904998518865, 4.43436977973458726166410884658, 5.029603964660056999268783372112, 7.314183581769516873960677364237, 7.913760251810141710878744527928, 8.864083910913874226688999536753, 9.30001649122132496716389013391, 10.42865847488305359026654108639, 11.62765650092598842370559297532, 12.28159740316235757527133061491, 13.13420767424933350111960419107, 14.180608685451632800307237067239, 15.2533461004311998804382745, 16.00282616692218414983986912339, 17.122869029656168128630680273433, 17.957564721277205524503021615884, 19.14656445507540218730448926516, 19.47452570338485443505210710311, 20.37090507750674613795700614464, 21.20306396018005154348746942338, 21.54424390537011023259251940117, 22.85103975217245383984773363548, 24.098896260148511476357757783435

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