Properties

Degree $2$
Conductor $3600$
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

Degree: \(2\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.584 + 0.811i$
Motivic weight: \(0\)
Character: $\chi_{3600} (757, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3600,\ (\ :0),\ 0.584 + 0.811i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.144775467\)
\(L(\frac12)\) \(\approx\) \(1.144775467\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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