Properties

Label 2-600-120.107-c0-0-1
Degree $2$
Conductor $600$
Sign $0.850 + 0.525i$
Analytic cond. $0.299439$
Root an. cond. $0.547210$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 0.707i)2-s + (−0.707 − 0.707i)3-s − 1.00i·4-s + 1.00·6-s + (0.707 + 0.707i)8-s + 1.00i·9-s + (−0.707 + 0.707i)12-s − 1.00·16-s + (1.41 − 1.41i)17-s + (−0.707 − 0.707i)18-s − 2i·19-s − 1.00i·24-s + (0.707 − 0.707i)27-s + (0.707 − 0.707i)32-s + 2.00i·34-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (−0.707 − 0.707i)3-s − 1.00i·4-s + 1.00·6-s + (0.707 + 0.707i)8-s + 1.00i·9-s + (−0.707 + 0.707i)12-s − 1.00·16-s + (1.41 − 1.41i)17-s + (−0.707 − 0.707i)18-s − 2i·19-s − 1.00i·24-s + (0.707 − 0.707i)27-s + (0.707 − 0.707i)32-s + 2.00i·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.850 + 0.525i$
Analytic conductor: \(0.299439\)
Root analytic conductor: \(0.547210\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :0),\ 0.850 + 0.525i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5135013168\)
\(L(\frac12)\) \(\approx\) \(0.5135013168\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 \)
good7 \( 1 - iT^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + (-1.41 + 1.41i)T - iT^{2} \)
19 \( 1 + 2iT - T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + (-1.41 - 1.41i)T + iT^{2} \)
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

−10.83807134104021538919445611166, −9.805113774244565345464745877110, −9.028749079948875148683035615096, −7.891029457169636612642187342351, −7.24158237298587336364111639163, −6.52609314592678570021388224505, −5.44616737155818135769791162795, −4.80159844471133660201479248304, −2.60587304832788196123186125509, −0.934443192434912603597736463747, 1.52492142328152317322830966531, 3.39863170052532323827461680497, 4.04886870152949275112666470514, 5.45707401313710585920192779646, 6.37184887812364498097616006368, 7.71542836175261784312940339271, 8.445017224444576913761884596745, 9.542691856045689371242719581915, 10.26284025884374299403228636652, 10.62300218863255284045641217800

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