↑

Properties

Label 1-407-407.156-r0-0-0
Degree $1$
Conductor $407$
Sign $-0.939 + 0.341i$
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

Related objects

Downloads

Learn more

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

\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(407\)    =    \(11 \cdot 37\)
Sign: $-0.939 + 0.341i$
Analytic conductor: \(1.89010\)
Root analytic conductor: \(1.89010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{407} (156, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 407,\ (0:\ ),\ -0.939 + 0.341i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2799265386 + 1.591044741i\)
\(L(\frac12)\) \(\approx\) \(0.2799265386 + 1.591044741i\)
\(L(1)\) \(\approx\) \(0.9596085914 + 0.9821870528i\)
\(L(1)\) \(\approx\) \(0.9596085914 + 0.9821870528i\)

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 \( 1 + (0.587 - 0.809i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.99508350928864498543736873006, −22.940364243760080994194528535906, −22.11095674433635870249823120484, −21.02103376658809166752413888350, −20.30276999742813673222616730673, −19.98739919510903446177160660466, −19.11560681744291588203561825527, −18.18981178419483453736066092055, −16.80679454093031845317790811606, −15.79941017907541644810614621059, −15.09019856180046775167429138286, −13.781265835987208016239541647168, −13.263172093242525160945679651145, −12.69495534535064426018067483707, −11.33235587104238406639841673718, −10.50895295173347614139205602144, −9.321767540105463692571819425404, −8.91294472987377640761023582660, −7.74535723585742300246218542788, −6.36460639782566668360446277105, −4.82689884654445758312924703560, −4.086855066871275417917055390388, −3.32161958492024241484590294608, −2.07122912041411329851004448938, −0.737055818783322320435620892842, 2.1459786239925103315520323247, 3.47667684515516324368217845221, 3.748456737035527933411140866103, 5.52308771526889331786239200691, 6.57514543212594430660552595706, 7.21188504952012521359419902532, 8.320142924358819093167705207197, 8.97829333258483657559841819254, 10.05278663122437036155367541886, 11.52809625213755457462359151928, 12.59372544738912939472693739407, 13.45285799940968232145149717235, 14.17961486523916359775224908792, 15.226865400928312155635678663704, 15.509720574728849727044063985635, 16.46501991935195909006957879364, 18.04866306702265918530643220255, 18.48016764406498036667875903115, 19.29559558743206495004408386818, 20.3885217852573469529675666478, 21.55475655530847375086032835363, 22.15452144821962863968974142087, 23.124442451472064391854655848070, 23.852178406837979860151139664105, 24.9271020436066728508222678285

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