↑

Properties

Label 2-30e2-5.2-c0-0-1
Degree $2$
Conductor $900$
Sign $0.437 + 0.899i$
Analytic cond. $0.449158$
Root an. cond. $0.670192$
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  + (−1.22 − 1.22i)7-s + (1.22 − 1.22i)13-s − i·19-s + 31-s + (−1.22 + 1.22i)43-s + 1.99i·49-s − 61-s + (1.22 + 1.22i)67-s − 2i·79-s − 2.99·91-s + (1.22 + 1.22i)97-s + i·109-s + ⋯
L(s)  = 1  + (−1.22 − 1.22i)7-s + (1.22 − 1.22i)13-s − i·19-s + 31-s + (−1.22 + 1.22i)43-s + 1.99i·49-s − 61-s + (1.22 + 1.22i)67-s − 2i·79-s − 2.99·91-s + (1.22 + 1.22i)97-s + i·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.437 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.437 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.437 + 0.899i$
Analytic conductor: \(0.449158\)
Root analytic conductor: \(0.670192\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (757, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :0),\ 0.437 + 0.899i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8715706983\)
\(L(\frac12)\) \(\approx\) \(0.8715706983\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (1.22 + 1.22i)T + iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (-1.22 + 1.22i)T - iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + iT - T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T + T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (1.22 - 1.22i)T - iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 + (-1.22 - 1.22i)T + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + 2iT - T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-1.22 - 1.22i)T + iT^{2} \)
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

−10.25353791641766692345523681673, −9.462176123197861403004556759528, −8.459720517777055832355086295048, −7.59355858488529286103753863904, −6.66124527685326685047630787162, −6.05536351102163369485977570454, −4.76477946321248790789473686286, −3.65813058030188529823592112323, −2.95675114766430724629736550549, −0.913355882118608624809784294802, 1.87344142490513574483838728023, 3.13224511547547583092011411899, 4.02866988328401533974858982616, 5.40589337535647924101725424261, 6.26353907226557983656036270293, 6.73802011082313426482765853299, 8.170308511243535767782982734029, 8.862893281419158531774215734369, 9.533012411108889551851840406790, 10.32117757414847022994671968457

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