↑

Properties

Label 2-1775-1775.354-c0-0-5
Degree $2$
Conductor $1775$
Sign $0.116 + 0.993i$
Analytic cond. $0.885840$
Root an. cond. $0.941190$
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  + (0.668 + 0.217i)2-s + (1.16 − 1.60i)3-s + (−0.409 − 0.297i)4-s + (0.393 + 0.919i)5-s + (1.12 − 0.818i)6-s + (−0.622 − 0.856i)8-s + (−0.904 − 2.78i)9-s + (0.0629 + 0.699i)10-s + (−0.954 + 0.309i)12-s + (1.93 + 0.441i)15-s + (−0.0734 − 0.226i)16-s − 2.05i·18-s + (−0.0725 + 0.0527i)19-s + (0.112 − 0.493i)20-s − 2.09·24-s + (−0.691 + 0.722i)25-s + ⋯
L(s)  = 1  + (0.668 + 0.217i)2-s + (1.16 − 1.60i)3-s + (−0.409 − 0.297i)4-s + (0.393 + 0.919i)5-s + (1.12 − 0.818i)6-s + (−0.622 − 0.856i)8-s + (−0.904 − 2.78i)9-s + (0.0629 + 0.699i)10-s + (−0.954 + 0.309i)12-s + (1.93 + 0.441i)15-s + (−0.0734 − 0.226i)16-s − 2.05i·18-s + (−0.0725 + 0.0527i)19-s + (0.112 − 0.493i)20-s − 2.09·24-s + (−0.691 + 0.722i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.116 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.116 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1775\)    =    \(5^{2} \cdot 71\)
Sign: $0.116 + 0.993i$
Analytic conductor: \(0.885840\)
Root analytic conductor: \(0.941190\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1775} (354, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1775,\ (\ :0),\ 0.116 + 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.038455503\)
\(L(\frac12)\) \(\approx\) \(2.038455503\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.393 - 0.919i)T \)
71 \( 1 + (-0.809 - 0.587i)T \)
good2 \( 1 + (-0.668 - 0.217i)T + (0.809 + 0.587i)T^{2} \)
3 \( 1 + (-1.16 + 1.60i)T + (-0.309 - 0.951i)T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + (0.809 + 0.587i)T^{2} \)
13 \( 1 + (-0.809 + 0.587i)T^{2} \)
17 \( 1 + (0.309 - 0.951i)T^{2} \)
19 \( 1 + (0.0725 - 0.0527i)T + (0.309 - 0.951i)T^{2} \)
23 \( 1 + (-0.809 - 0.587i)T^{2} \)
29 \( 1 + (-0.766 - 0.557i)T + (0.309 + 0.951i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (-1.90 + 0.617i)T + (0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.809 - 0.587i)T^{2} \)
43 \( 1 - 1.02iT - T^{2} \)
47 \( 1 + (0.309 + 0.951i)T^{2} \)
53 \( 1 + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (0.809 + 0.587i)T^{2} \)
67 \( 1 + (0.309 - 0.951i)T^{2} \)
73 \( 1 + (0.975 + 0.316i)T + (0.809 + 0.587i)T^{2} \)
79 \( 1 + (-1.21 - 0.885i)T + (0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.919 - 1.26i)T + (-0.309 + 0.951i)T^{2} \)
89 \( 1 + (-0.0829 + 0.255i)T + (-0.809 - 0.587i)T^{2} \)
97 \( 1 + (0.309 + 0.951i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.332993976872976450953431958075, −8.298946037542260556210433823656, −7.65692549276985564401967279371, −6.67384417405515201167230539503, −6.42377616807603731183296480489, −5.52092279877874876501027902504, −4.08096778835079413036654325199, −3.16154090176663358724420142978, −2.46238818436259807107078234955, −1.22729632063598389830377057384, 2.21809340137070887865732256590, 3.06554017331612415508757453916, 3.97388984521536670470107802259, 4.58619256475348329563458610035, 5.10090881413982280019670278336, 6.00421744649538840297217158255, 7.84573419568800604500675613451, 8.301572763119310568218338834311, 9.002892067306468150775500890611, 9.526652170896956395857156656182

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