↑

Properties

Label 2-675-5.2-c0-0-2
Degree $2$
Conductor $675$
Sign $-0.229 - 0.973i$
Analytic cond. $0.336868$
Root an. cond. $0.580404$
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)2-s + 1.99i·4-s + (−1.22 + 1.22i)8-s − 0.999·16-s + (−1.22 − 1.22i)17-s − i·19-s + (−1.22 + 1.22i)23-s + 31-s − 2.99i·34-s + (1.22 − 1.22i)38-s − 2.99·46-s − i·49-s + (1.22 − 1.22i)53-s − 61-s + (1.22 + 1.22i)62-s + ⋯
L(s)  = 1  + (1.22 + 1.22i)2-s + 1.99i·4-s + (−1.22 + 1.22i)8-s − 0.999·16-s + (−1.22 − 1.22i)17-s − i·19-s + (−1.22 + 1.22i)23-s + 31-s − 2.99i·34-s + (1.22 − 1.22i)38-s − 2.99·46-s − i·49-s + (1.22 − 1.22i)53-s − 61-s + (1.22 + 1.22i)62-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-0.229 - 0.973i$
Analytic conductor: \(0.336868\)
Root analytic conductor: \(0.580404\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :0),\ -0.229 - 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.654310070\)
\(L(\frac12)\) \(\approx\) \(1.654310070\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (-1.22 - 1.22i)T + iT^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + (1.22 + 1.22i)T + iT^{2} \)
19 \( 1 + iT - T^{2} \)
23 \( 1 + (1.22 - 1.22i)T - iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T + T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (-1.22 + 1.22i)T - iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 - iT - T^{2} \)
83 \( 1 + (1.22 - 1.22i)T - iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + iT^{2} \)
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

−11.36824324813473996698183267678, −9.983639758045107162979528361938, −8.962804749152519098262569767721, −8.033732683895072741146678126470, −7.12526193066663347468883255759, −6.54472225855985923440408992127, −5.49283572481404778194727979567, −4.73605974514350794377128674745, −3.82453013013801892490597477474, −2.57668953878948128329649032184, 1.71008018474612900358493498695, 2.73446325593226330464990210468, 4.01435693347454554515410215232, 4.49702805778516744208100153621, 5.83439635965169869135240927353, 6.39599223232646567631386509069, 7.943234816988079955878296768095, 8.919158758895878612569809553597, 10.22537592441517369642854860081, 10.48306131269439949108801901591

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