↑

Properties

Label 2-1775-1775.141-c0-0-3
Degree $2$
Conductor $1775$
Sign $-0.953 - 0.300i$
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  + (1.55 + 1.13i)2-s + (0.292 + 0.901i)3-s + (0.839 + 2.58i)4-s + (−0.995 − 0.0896i)5-s + (−0.564 + 1.73i)6-s + (−1.02 + 3.14i)8-s + (0.0823 − 0.0598i)9-s + (−1.45 − 1.26i)10-s + (−2.08 + 1.51i)12-s + (−0.210 − 0.923i)15-s + (−2.96 + 2.15i)16-s + 0.196·18-s + (0.578 − 1.78i)19-s + (−0.604 − 2.64i)20-s − 3.13·24-s + (0.983 + 0.178i)25-s + ⋯
L(s)  = 1  + (1.55 + 1.13i)2-s + (0.292 + 0.901i)3-s + (0.839 + 2.58i)4-s + (−0.995 − 0.0896i)5-s + (−0.564 + 1.73i)6-s + (−1.02 + 3.14i)8-s + (0.0823 − 0.0598i)9-s + (−1.45 − 1.26i)10-s + (−2.08 + 1.51i)12-s + (−0.210 − 0.923i)15-s + (−2.96 + 2.15i)16-s + 0.196·18-s + (0.578 − 1.78i)19-s + (−0.604 − 2.64i)20-s − 3.13·24-s + (0.983 + 0.178i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 - 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 - 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1775\)    =    \(5^{2} \cdot 71\)
Sign: $-0.953 - 0.300i$
Analytic conductor: \(0.885840\)
Root analytic conductor: \(0.941190\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1775} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1775,\ (\ :0),\ -0.953 - 0.300i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.511221805\)
\(L(\frac12)\) \(\approx\) \(2.511221805\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.995 + 0.0896i)T \)
71 \( 1 + (-0.309 - 0.951i)T \)
good2 \( 1 + (-1.55 - 1.13i)T + (0.309 + 0.951i)T^{2} \)
3 \( 1 + (-0.292 - 0.901i)T + (-0.809 + 0.587i)T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (-0.309 + 0.951i)T^{2} \)
17 \( 1 + (0.809 + 0.587i)T^{2} \)
19 \( 1 + (-0.578 + 1.78i)T + (-0.809 - 0.587i)T^{2} \)
23 \( 1 + (-0.309 - 0.951i)T^{2} \)
29 \( 1 + (0.427 + 1.31i)T + (-0.809 + 0.587i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (1.51 - 1.10i)T + (0.309 - 0.951i)T^{2} \)
41 \( 1 + (-0.309 + 0.951i)T^{2} \)
43 \( 1 + 0.786T + T^{2} \)
47 \( 1 + (0.809 - 0.587i)T^{2} \)
53 \( 1 + (0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (-0.309 - 0.951i)T^{2} \)
67 \( 1 + (0.809 + 0.587i)T^{2} \)
73 \( 1 + (-0.635 - 0.462i)T + (0.309 + 0.951i)T^{2} \)
79 \( 1 + (-0.530 - 1.63i)T + (-0.809 + 0.587i)T^{2} \)
83 \( 1 + (-0.385 + 1.18i)T + (-0.809 - 0.587i)T^{2} \)
89 \( 1 + (0.766 + 0.557i)T + (0.309 + 0.951i)T^{2} \)
97 \( 1 + (0.809 - 0.587i)T^{2} \)
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

−9.636341437040091965334609584691, −8.699619185408099097241931933606, −8.130666619434249944742270658048, −7.08193220339183270347491315497, −6.82745291610722716847730494275, −5.51786399277036836897158784738, −4.79784226360483010077897726373, −4.23432629263312095288525132673, −3.50274251079765084103943443216, −2.76838997246062345139827644975, 1.28295828983045522161693004944, 2.15040906770554571148046412023, 3.34916538156164384473322554428, 3.79507221223305958768152998331, 4.85979610428677331870505360555, 5.63032774172782728011896113811, 6.66864744852341059684088353483, 7.30410093540369789664189753808, 8.166637162227362277378751331442, 9.312630889675622751478608763793

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