↑

Properties

Label 1-407-407.141-r0-0-0
Degree $1$
Conductor $407$
Sign $0.767 - 0.640i$
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  + (0.997 − 0.0697i)2-s + (0.848 − 0.529i)3-s + (0.990 − 0.139i)4-s + (−0.559 + 0.829i)5-s + (0.809 − 0.587i)6-s + (−0.615 − 0.788i)7-s + (0.978 − 0.207i)8-s + (0.438 − 0.898i)9-s + (−0.5 + 0.866i)10-s + (0.766 − 0.642i)12-s + (0.719 − 0.694i)13-s + (−0.669 − 0.743i)14-s + (−0.0348 + 0.999i)15-s + (0.961 − 0.275i)16-s + (0.719 + 0.694i)17-s + (0.374 − 0.927i)18-s + ⋯
L(s)  = 1  + (0.997 − 0.0697i)2-s + (0.848 − 0.529i)3-s + (0.990 − 0.139i)4-s + (−0.559 + 0.829i)5-s + (0.809 − 0.587i)6-s + (−0.615 − 0.788i)7-s + (0.978 − 0.207i)8-s + (0.438 − 0.898i)9-s + (−0.5 + 0.866i)10-s + (0.766 − 0.642i)12-s + (0.719 − 0.694i)13-s + (−0.669 − 0.743i)14-s + (−0.0348 + 0.999i)15-s + (0.961 − 0.275i)16-s + (0.719 + 0.694i)17-s + (0.374 − 0.927i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.767 - 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.767 - 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(407\)    =    \(11 \cdot 37\)
Sign: $0.767 - 0.640i$
Analytic conductor: \(1.89010\)
Root analytic conductor: \(1.89010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{407} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 407,\ (0:\ ),\ 0.767 - 0.640i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.778754857 - 1.007338069i\)
\(L(\frac12)\) \(\approx\) \(2.778754857 - 1.007338069i\)
\(L(1)\) \(\approx\) \(2.168818533 - 0.4752553710i\)
\(L(1)\) \(\approx\) \(2.168818533 - 0.4752553710i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
37 \( 1 \)
good2 \( 1 + (0.997 - 0.0697i)T \)
3 \( 1 + (0.848 - 0.529i)T \)
5 \( 1 + (-0.559 + 0.829i)T \)
7 \( 1 + (-0.615 - 0.788i)T \)
13 \( 1 + (0.719 - 0.694i)T \)
17 \( 1 + (0.719 + 0.694i)T \)
19 \( 1 + (-0.848 + 0.529i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (-0.669 + 0.743i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
41 \( 1 + (-0.882 - 0.469i)T \)
43 \( 1 - T \)
47 \( 1 + (0.669 + 0.743i)T \)
53 \( 1 + (0.559 + 0.829i)T \)
59 \( 1 + (-0.0348 + 0.999i)T \)
61 \( 1 + (-0.438 - 0.898i)T \)
67 \( 1 + (-0.939 + 0.342i)T \)
71 \( 1 + (-0.997 - 0.0697i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (-0.961 - 0.275i)T \)
83 \( 1 + (-0.719 - 0.694i)T \)
89 \( 1 + (0.939 + 0.342i)T \)
97 \( 1 + (-0.913 + 0.406i)T \)
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

−24.44628595825842196160862995384, −23.514853155706242093059856390109, −22.70877451219772923559927819478, −21.59794885237191024594381841255, −21.10027064270315002607251096811, −20.32574741483599787501497257139, −19.393483153827098945417325975209, −18.83965722334095574677463748969, −16.84743068266870421854292563822, −16.239712109331597883542555440278, −15.4035284400282953908652503393, −14.97273128911035705102999346148, −13.55256269205123746093977111460, −13.210146736361829664224726712149, −12.017930626780507802413740314880, −11.33026999778687130113795574293, −9.897222682076927347475173059556, −8.952472227654595498386930974261, −8.10622172989869834732570881650, −6.97883871369655752539199065515, −5.68587459098008567997741423529, −4.73952060482465926027959254676, −3.832664675527357225341532729057, −2.98195368312937716303780598331, −1.7773008228065655711706635325, 1.34716240196885194218465951794, 2.81877083400667225464281417015, 3.46592281892810472749095690415, 4.21471867759716762268427380658, 6.016725022556351902325845555503, 6.769529985384743691984916842522, 7.56119275655336332773126949844, 8.48137744652972827526125689182, 10.26275561848538055633914249967, 10.67095271782548345453694099252, 12.08156766529875086309660735961, 12.827203523316393298057659657218, 13.60420102715734629504112661653, 14.49710946000252011571214150979, 15.07399839309570805627341540243, 15.98302997233872347792495227479, 17.0774978009699705117769206243, 18.58820764744890422602007277565, 19.15284213292787921964425689643, 20.018492659576322811479030104477, 20.64036110094121405654863767944, 21.65334924682055708247393857524, 22.82947353485820202597458974223, 23.259604918035712925072466516890, 23.964832778669933794103127064585

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