Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \cdot 5^{2} $
Sign $0.998 - 0.0618i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + i)13-s + (1.41 − 1.41i)17-s + 1.41·29-s + (1 + i)37-s − 1.41i·41-s + i·49-s + (1.41 + 1.41i)53-s + (1 − i)73-s − 1.41·89-s + (1 + i)97-s − 1.41i·101-s + ⋯
L(s)  = 1  + (−1 + i)13-s + (1.41 − 1.41i)17-s + 1.41·29-s + (1 + i)37-s − 1.41i·41-s + i·49-s + (1.41 + 1.41i)53-s + (1 − i)73-s − 1.41·89-s + (1 + i)97-s − 1.41i·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
\( \varepsilon \)  =  $0.998 - 0.0618i$
motivic weight  =  \(0\)
character  :  $\chi_{3600} (1007, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 3600,\ (\ :0),\ 0.998 - 0.0618i)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(1.289309924\)
\(L(\frac12)\)  \(\approx\)  \(1.289309924\)
\(L(1)\)   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 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (1 - i)T - iT^{2} \)
17 \( 1 + (-1.41 + 1.41i)T - iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 - 1.41T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-1 - i)T + iT^{2} \)
41 \( 1 + 1.41iT - T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (-1.41 - 1.41i)T + iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-1 + i)T - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + 1.41T + T^{2} \)
97 \( 1 + (-1 - i)T + iT^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.834183045858896215590023219831, −7.85867063911802813772846974296, −7.30081693918475434795623116413, −6.63065872678524702680547266745, −5.67988046107564714656975990578, −4.90449726426965376192210600265, −4.25824456212995958992485879422, −3.08710023255538321158036724516, −2.39260388633025422999209861023, −1.03496738874621367978271446123, 1.00610473011614275017122542910, 2.31431995466167475123812049611, 3.20289258474364028574680896810, 4.04770330587375452454865044459, 5.05493350352688407288564698729, 5.66979806296969947094178306581, 6.45791472496820676369570347129, 7.35541345586205908549329858971, 8.077832017830271848652094282791, 8.463188702016049137277626859858

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