Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \cdot 5^{2} $
Sign $0.584 - 0.811i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
\( \varepsilon \)  =  $0.584 - 0.811i$
motivic weight  =  \(0\)
character  :  $\chi_{3600} (1693, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 3600,\ (\ :0),\ 0.584 - 0.811i)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(1.144775467\)
\(L(\frac12)\)  \(\approx\)  \(1.144775467\)
\(L(1)\)   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 \( 1 - iT \)
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 \( 1 - 2T + T^{2} \)
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 \( 1 - 2iT - T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - iT^{2} \)
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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

−8.526233545765959870534761029644, −8.237971496252668051365799766698, −7.09624950196721262391253256446, −6.89490832464134106515289714728, −5.74601958566405836817432535151, −5.32900669016639210065358322191, −4.34747779460662295910268534792, −3.63399016935119989243603171591, −2.48480195480771585385219921642, −0.898819031395489928031866056968, 1.03148092965203080579482246530, 2.10842343207064511430585286330, 3.05542926507876727353684843455, 3.83710651073430088809657606669, 4.65177911743131635148143819386, 5.46057489774343533994656622524, 6.22559801106408865588512433282, 7.32857858289337515784577252081, 8.012384872896461317751539076957, 8.924840331713877732469922041288

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