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: $\chi_{3600} (1693, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3600,\ (\ :0),\ 0.584 - 0.811i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5684665075\)
\(L(\frac12)\) \(\approx\) \(0.5684665075\)
\(L(1)\) 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

−8.840329471585470289009577905146, −8.307954698974186515567994325747, −7.64546647484322424397325663200, −6.40424146961632750196400319984, −5.83733642954096286456462120135, −4.80637539491969613091654765738, −4.04885870075158876883996776906, −3.43220991952788035647079883045, −2.21963929158248161733974633278, −1.56324844890515915473163182591, 0.32011811031841551413749580584, 2.09847707073508280553719848146, 3.21988968857554125625413749981, 4.43682765824532894580321547985, 4.69870004726746144078376948285, 5.75072337935010036189406773612, 6.61661409666853980558975589704, 6.88925762661635178074584972853, 7.950447178398915132708878322573, 8.464483206693136692419725031041

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