Properties

Degree $2$
Conductor $3600$
Sign $0.987 - 0.160i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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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

Degree: \(2\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.987 - 0.160i$
Motivic weight: \(0\)
Character: $\chi_{3600} (3493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3600,\ (\ :0),\ 0.987 - 0.160i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8473083147\)
\(L(\frac12)\) \(\approx\) \(0.8473083147\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

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

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