Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \cdot 5^{2} $
Sign $0.987 + 0.160i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 8-s + 16-s + (1 + i)17-s + (1 − i)19-s + (−1 − i)23-s − 32-s + (−1 − i)34-s + (−1 + i)38-s + (1 + i)46-s + (1 + i)47-s + i·49-s − 2i·53-s + (1 + i)61-s + ⋯
L(s)  = 1  − 2-s + 4-s − 8-s + 16-s + (1 + i)17-s + (1 − i)19-s + (−1 − i)23-s − 32-s + (−1 − i)34-s + (−1 + i)38-s + (1 + i)46-s + (1 + i)47-s + i·49-s − 2i·53-s + (1 + i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
\( \varepsilon \)  =  $0.987 + 0.160i$
motivic weight  =  \(0\)
character  :  $\chi_{3600} (2557, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 3600,\ (\ :0),\ 0.987 + 0.160i)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.8473083147\)
\(L(\frac12)\)  \(\approx\)  \(0.8473083147\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - iT^{2} \)
11 \( 1 - iT^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (-1 - i)T + iT^{2} \)
19 \( 1 + (-1 + i)T - iT^{2} \)
23 \( 1 + (1 + i)T + iT^{2} \)
29 \( 1 - iT^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-1 - i)T + iT^{2} \)
53 \( 1 + 2iT - T^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + (-1 - i)T + iT^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + 2iT - T^{2} \)
83 \( 1 - 2T + T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + iT^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.747506268940613382712887968863, −7.949276505332593696836538162918, −7.49948590138733128515487477591, −6.55877104087657494037607875114, −5.96427359175341310717797828612, −5.08251563771190697598871274012, −3.90029188327220822394473588323, −2.99415304506268720952436637574, −2.05762052439398049788431617748, −0.901534009153222065044194060008, 1.00039767115762042892346774083, 2.05766774731570020841204835253, 3.12711639803258165308250889241, 3.85240628417037585804803315095, 5.33240190214868689701780887518, 5.73867384980883470554502285022, 6.75788248891037997492784621781, 7.54005616866590754321907977609, 7.896773680884115952924095968678, 8.763907658950581860575461707022

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