Properties

Degree $2$
Conductor $3600$
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

Degree: \(2\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.998 + 0.0618i$
Motivic weight: \(0\)
Character: $\chi_{3600} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3600,\ (\ :0),\ 0.998 + 0.0618i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.289309924\)
\(L(\frac12)\) \(\approx\) \(1.289309924\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

Imaginary part of the first few zeros on the critical line

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

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