↑

Properties

Label 1-407-407.9-r0-0-0
Degree $1$
Conductor $407$
Sign $-0.841 - 0.539i$
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  + (0.961 + 0.275i)2-s + (−0.615 + 0.788i)3-s + (0.848 + 0.529i)4-s + (−0.719 − 0.694i)5-s + (−0.809 + 0.587i)6-s + (−0.882 + 0.469i)7-s + (0.669 + 0.743i)8-s + (−0.241 − 0.970i)9-s + (−0.5 − 0.866i)10-s + (−0.939 + 0.342i)12-s + (−0.997 + 0.0697i)13-s + (−0.978 + 0.207i)14-s + (0.990 − 0.139i)15-s + (0.438 + 0.898i)16-s + (−0.997 − 0.0697i)17-s + (0.0348 − 0.999i)18-s + ⋯
L(s)  = 1  + (0.961 + 0.275i)2-s + (−0.615 + 0.788i)3-s + (0.848 + 0.529i)4-s + (−0.719 − 0.694i)5-s + (−0.809 + 0.587i)6-s + (−0.882 + 0.469i)7-s + (0.669 + 0.743i)8-s + (−0.241 − 0.970i)9-s + (−0.5 − 0.866i)10-s + (−0.939 + 0.342i)12-s + (−0.997 + 0.0697i)13-s + (−0.978 + 0.207i)14-s + (0.990 − 0.139i)15-s + (0.438 + 0.898i)16-s + (−0.997 − 0.0697i)17-s + (0.0348 − 0.999i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.841 - 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.841 - 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(407\)    =    \(11 \cdot 37\)
Sign: $-0.841 - 0.539i$
Analytic conductor: \(1.89010\)
Root analytic conductor: \(1.89010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{407} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 407,\ (0:\ ),\ -0.841 - 0.539i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.08500773585 + 0.2900653380i\)
\(L(\frac12)\) \(\approx\) \(-0.08500773585 + 0.2900653380i\)
\(L(1)\) \(\approx\) \(0.7953986798 + 0.3760163362i\)
\(L(1)\) \(\approx\) \(0.7953986798 + 0.3760163362i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
37 \( 1 \)
good2 \( 1 + (0.961 + 0.275i)T \)
3 \( 1 + (-0.615 + 0.788i)T \)
5 \( 1 + (-0.719 - 0.694i)T \)
7 \( 1 + (-0.882 + 0.469i)T \)
13 \( 1 + (-0.997 + 0.0697i)T \)
17 \( 1 + (-0.997 - 0.0697i)T \)
19 \( 1 + (-0.615 + 0.788i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.978 - 0.207i)T \)
31 \( 1 + (-0.809 - 0.587i)T \)
41 \( 1 + (-0.374 - 0.927i)T \)
43 \( 1 + T \)
47 \( 1 + (-0.978 + 0.207i)T \)
53 \( 1 + (-0.719 + 0.694i)T \)
59 \( 1 + (0.990 - 0.139i)T \)
61 \( 1 + (-0.241 + 0.970i)T \)
67 \( 1 + (0.173 + 0.984i)T \)
71 \( 1 + (0.961 - 0.275i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (0.438 - 0.898i)T \)
83 \( 1 + (-0.997 - 0.0697i)T \)
89 \( 1 + (0.173 - 0.984i)T \)
97 \( 1 + (-0.104 + 0.994i)T \)
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

−23.79074301681006759854238914003, −22.85777968317978669459011567335, −22.30600823006120703497996058194, −21.700024126682070518091497573592, −19.857710153937055139791528709038, −19.75252973324958833617818560030, −18.88382388292684181396321001880, −17.74775443884006154211713129696, −16.65519326861196407466020218524, −15.78777508481074533472515123633, −14.89554756572178614564883899070, −13.867794102161589357052093712819, −12.9916700281825787224521804431, −12.3834702804570667641346419479, −11.30272121587059789776294883909, −10.84270306359994224505307036291, −9.681417510804884838071760136701, −7.80036239016931218573728987083, −6.92202961231548458942752894247, −6.51025039498279527607322354861, −5.22642154047775742230536604904, −4.12198631737652510831197875413, −3.02073527992484492976517451129, −1.98870813310274216646015702839, −0.12413555354410348094885069333, 2.34633495661948739492647364306, 3.71129861625194740472546065179, 4.3414960152782749915813247149, 5.33092049813173955288019652694, 6.18035437002611797297152738747, 7.22420099723967233475963306369, 8.52724263647280688236808404017, 9.53856073580319791765117059577, 10.73237378473042385193044023876, 11.70805868588919011875648051507, 12.43098112600754060087884827042, 13.0026368197564234841256686715, 14.556872444800381540239227430236, 15.250180550712662976473254413146, 16.03449562214623667961501587454, 16.59053929560159865483269381489, 17.36488635795986565141160916035, 18.935737936272612051745434201015, 20.00481850032860303656670549048, 20.620732341183788402524750048019, 21.64095983279526937664023679257, 22.450064565835230458692919006219, 22.803766441708507346900641049153, 23.97963294286346011042197328633, 24.44499632910525858614534778426

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