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Properties

Label 1-407-407.288-r0-0-0
Degree $1$
Conductor $407$
Sign $-0.491 - 0.871i$
Analytic cond. $1.89010$
Root an. cond. $1.89010$
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  + (−0.406 − 0.913i)2-s + (0.978 − 0.207i)3-s + (−0.669 + 0.743i)4-s + (0.406 − 0.913i)5-s + (−0.587 − 0.809i)6-s + (−0.669 + 0.743i)7-s + (0.951 + 0.309i)8-s + (0.913 − 0.406i)9-s − 10-s + (−0.5 + 0.866i)12-s + (−0.994 − 0.104i)13-s + (0.951 + 0.309i)14-s + (0.207 − 0.978i)15-s + (−0.104 − 0.994i)16-s + (0.994 − 0.104i)17-s + (−0.743 − 0.669i)18-s + ⋯
L(s)  = 1  + (−0.406 − 0.913i)2-s + (0.978 − 0.207i)3-s + (−0.669 + 0.743i)4-s + (0.406 − 0.913i)5-s + (−0.587 − 0.809i)6-s + (−0.669 + 0.743i)7-s + (0.951 + 0.309i)8-s + (0.913 − 0.406i)9-s − 10-s + (−0.5 + 0.866i)12-s + (−0.994 − 0.104i)13-s + (0.951 + 0.309i)14-s + (0.207 − 0.978i)15-s + (−0.104 − 0.994i)16-s + (0.994 − 0.104i)17-s + (−0.743 − 0.669i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.491 - 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.491 - 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(407\)    =    \(11 \cdot 37\)
Sign: $-0.491 - 0.871i$
Analytic conductor: \(1.89010\)
Root analytic conductor: \(1.89010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{407} (288, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 407,\ (0:\ ),\ -0.491 - 0.871i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6840224717 - 1.171014095i\)
\(L(\frac12)\) \(\approx\) \(0.6840224717 - 1.171014095i\)
\(L(1)\) \(\approx\) \(0.9055437817 - 0.6715705403i\)
\(L(1)\) \(\approx\) \(0.9055437817 - 0.6715705403i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
37 \( 1 \)
good2 \( 1 + (-0.406 - 0.913i)T \)
3 \( 1 + (0.978 - 0.207i)T \)
5 \( 1 + (0.406 - 0.913i)T \)
7 \( 1 + (-0.669 + 0.743i)T \)
13 \( 1 + (-0.994 - 0.104i)T \)
17 \( 1 + (0.994 - 0.104i)T \)
19 \( 1 + (-0.207 - 0.978i)T \)
23 \( 1 - iT \)
29 \( 1 + (0.951 - 0.309i)T \)
31 \( 1 + (0.587 - 0.809i)T \)
41 \( 1 + (-0.978 + 0.207i)T \)
43 \( 1 + iT \)
47 \( 1 + (0.309 - 0.951i)T \)
53 \( 1 + (0.913 - 0.406i)T \)
59 \( 1 + (-0.207 + 0.978i)T \)
61 \( 1 + (0.406 - 0.913i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (0.913 + 0.406i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (-0.994 - 0.104i)T \)
83 \( 1 + (0.104 + 0.994i)T \)
89 \( 1 + (-0.866 - 0.5i)T \)
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

−25.06791165292557624553577776318, −23.80845071327317556220106214346, −23.029070989244863189171961296636, −22.11540657505709569532257240717, −21.24079882025091011233544581850, −19.94184595993631433993705287832, −19.23160737341446469440575141971, −18.68118130999142590171380225690, −17.50914132357724720234940087414, −16.70418365096749039819664846608, −15.73325650134870577443908815962, −14.88396544971922966359296542699, −14.08988725714853187159933192798, −13.70189658824809549177237291090, −12.41372590778300932926636648527, −10.46577257973748831541658488178, −10.07949267249969251978365009682, −9.34281819312006741719653436891, −8.05182064134788721786517993846, −7.29968551068315287031225405592, −6.59309488577200516764804160850, −5.32709431198526845448569034607, −3.97538303826515533589224370794, −3.025797564144587320497764281970, −1.558873019187627064172682013689, 0.86445000048359800672386888403, 2.289306277914192963201605640333, 2.82226455761612984951314100082, 4.18681639595578914478935961315, 5.21060659673024378726630225382, 6.80786271277529594374701721379, 8.13299146459543279880422553832, 8.70656417658302142229891772493, 9.70073281669829571381183574669, 10.01730480455609507989000365062, 11.832749411668909150835132940518, 12.50616207591431808603177555448, 13.128403684472216459630860135673, 14.01839160303773176817697160238, 15.154745359627615523803136749997, 16.29830440069155597833293033253, 17.17688533359799367399162337218, 18.224086535504468468144141997835, 19.04621208128118786721709571808, 19.73945894609669200824142249441, 20.402190843242565056774113871364, 21.37217001930732647321197897245, 21.80066571777098535036940900003, 23.02013562783111384550902193798, 24.351549689283761867390018200350

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