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Properties

Label 1-451-451.432-r1-0-0
Degree $1$
Conductor $451$
Sign $0.399 + 0.916i$
Analytic cond. $48.4666$
Root an. cond. $48.4666$
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 + (−0.156 + 0.987i)3-s + (−0.309 + 0.951i)4-s + (0.587 + 0.809i)5-s + (0.891 − 0.453i)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 + (−0.891 − 0.453i)12-s + (−0.156 + 0.987i)13-s + (−0.987 − 0.156i)14-s + (−0.891 + 0.453i)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 + (−0.156 + 0.987i)3-s + (−0.309 + 0.951i)4-s + (0.587 + 0.809i)5-s + (0.891 − 0.453i)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 + (−0.891 − 0.453i)12-s + (−0.156 + 0.987i)13-s + (−0.987 − 0.156i)14-s + (−0.891 + 0.453i)15-s + (−0.809 − 0.587i)16-s + (0.707 + 0.707i)17-s + (0.309 + 0.951i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 451 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.399 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 451 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.399 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(451\)    =    \(11 \cdot 41\)
Sign: $0.399 + 0.916i$
Analytic conductor: \(48.4666\)
Root analytic conductor: \(48.4666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{451} (432, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 451,\ (1:\ ),\ 0.399 + 0.916i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.407249187 + 0.9213992767i\)
\(L(\frac12)\) \(\approx\) \(1.407249187 + 0.9213992767i\)
\(L(1)\) \(\approx\) \(0.9460828648 + 0.1774552073i\)
\(L(1)\) \(\approx\) \(0.9460828648 + 0.1774552073i\)

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 \( 1 + (-0.156 + 0.987i)T \)
5 \( 1 + (0.587 + 0.809i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
13 \( 1 + (-0.156 + 0.987i)T \)
17 \( 1 + (0.707 + 0.707i)T \)
19 \( 1 + (0.891 - 0.453i)T \)
23 \( 1 + (0.809 + 0.587i)T \)
29 \( 1 + (-0.453 - 0.891i)T \)
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 \( 1 + (-0.453 + 0.891i)T \)
71 \( 1 + (0.707 - 0.707i)T \)
73 \( 1 + (0.587 - 0.809i)T \)
79 \( 1 + (0.453 + 0.891i)T \)
83 \( 1 + (0.309 + 0.951i)T \)
89 \( 1 + (-0.156 + 0.987i)T \)
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

−24.04286839402422471878742580675, −22.99810439475577687270359701640, −22.218213431670014815276021595622, −20.732430402040831124766528273859, −20.16401012870365787680559479614, −18.90288882823582131689414093890, −18.3022682709264489014932118120, −17.61572284324249156836842480342, −16.913193077060308074336315257181, −16.039154568920767276712259750310, −14.84951335368883497331847301350, −14.10592130020509235299825585941, −13.17950881051936930980757713509, −12.29584105740288543760799352621, −11.29305659110137567291489529023, −10.017958629384709729719479762689, −9.039707669718446834593031661036, −8.19355353159179267820900285350, −7.58918119075671418076311992293, −6.34929045864766181928295951156, −5.39898615454093979795083168067, −5.03576913092522852884854156636, −2.66544291675608996637924281099, −1.43510897286287427673882083556, −0.68639148848050594051825750588, 1.09335697591915836249989587191, 2.38501150744412969922747584220, 3.48683954909310011010346373015, 4.32849199643925239062083183032, 5.48204964182582192857888991668, 6.93202135613493115083881176080, 7.90258318110485289934177211234, 9.1947649373980479032075234165, 9.765662882735429369614936258044, 10.73345972711458408652335072102, 11.156846721095430216449738537935, 12.09928876655883456396721681602, 13.66169271792346434084735267426, 14.16651185084982574946218013893, 15.23950758889327737217150348757, 16.46021820420799657749540966631, 17.27031457107656794694768620628, 17.69712536039235219328035025563, 18.883093568927838534201853294180, 19.65397314317205102325290303807, 20.84370291399966329837192448168, 21.16998979634280927384652646671, 21.9501672950900435945205168322, 22.75051540075180681460181321781, 23.66477241022165724467025248876

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