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Properties

Label 2-600-120.53-c1-0-1
Degree $2$
Conductor $600$
Sign $-0.654 + 0.755i$
Analytic cond. $4.79102$
Root an. cond. $2.18884$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.864 + 1.11i)2-s + (−1.72 − 0.170i)3-s + (−0.506 + 1.93i)4-s + (−1.29 − 2.07i)6-s + (−2.06 + 2.06i)7-s + (−2.60 + 1.10i)8-s + (2.94 + 0.586i)9-s − 0.510·11-s + (1.20 − 3.24i)12-s + (0.750 − 0.750i)13-s + (−4.10 − 0.528i)14-s + (−3.48 − 1.95i)16-s + (−3.14 − 3.14i)17-s + (1.88 + 3.80i)18-s − 6.01·19-s + ⋯
L(s)  = 1  + (0.611 + 0.791i)2-s + (−0.995 − 0.0982i)3-s + (−0.253 + 0.967i)4-s + (−0.530 − 0.847i)6-s + (−0.782 + 0.782i)7-s + (−0.920 + 0.390i)8-s + (0.980 + 0.195i)9-s − 0.153·11-s + (0.346 − 0.937i)12-s + (0.208 − 0.208i)13-s + (−1.09 − 0.141i)14-s + (−0.871 − 0.489i)16-s + (−0.763 − 0.763i)17-s + (0.444 + 0.895i)18-s − 1.37·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 + 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.654 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $-0.654 + 0.755i$
Analytic conductor: \(4.79102\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ -0.654 + 0.755i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.140066 - 0.306683i\)
\(L(\frac12)\) \(\approx\) \(0.140066 - 0.306683i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.864 - 1.11i)T \)
3 \( 1 + (1.72 + 0.170i)T \)
5 \( 1 \)
good7 \( 1 + (2.06 - 2.06i)T - 7iT^{2} \)
11 \( 1 + 0.510T + 11T^{2} \)
13 \( 1 + (-0.750 + 0.750i)T - 13iT^{2} \)
17 \( 1 + (3.14 + 3.14i)T + 17iT^{2} \)
19 \( 1 + 6.01T + 19T^{2} \)
23 \( 1 + (-2.54 + 2.54i)T - 23iT^{2} \)
29 \( 1 - 5.10iT - 29T^{2} \)
31 \( 1 + 4.56T + 31T^{2} \)
37 \( 1 + (6.76 + 6.76i)T + 37iT^{2} \)
41 \( 1 + 4.24iT - 41T^{2} \)
43 \( 1 + (5.95 - 5.95i)T - 43iT^{2} \)
47 \( 1 + (-3.33 - 3.33i)T + 47iT^{2} \)
53 \( 1 + (-5.75 - 5.75i)T + 53iT^{2} \)
59 \( 1 - 1.16iT - 59T^{2} \)
61 \( 1 - 4.92iT - 61T^{2} \)
67 \( 1 + (-7.98 - 7.98i)T + 67iT^{2} \)
71 \( 1 + 5.09iT - 71T^{2} \)
73 \( 1 + (-3.20 - 3.20i)T + 73iT^{2} \)
79 \( 1 - 7.31iT - 79T^{2} \)
83 \( 1 + (4.77 + 4.77i)T + 83iT^{2} \)
89 \( 1 + 12.6T + 89T^{2} \)
97 \( 1 + (10.8 - 10.8i)T - 97iT^{2} \)
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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

−11.32378879003486162332318570913, −10.55186059459035905920946522580, −9.248835364074469154057059266315, −8.594256638267675011483720422286, −7.21938949366599231460233993378, −6.63104347637353830695572123669, −5.78938374093471834918102451008, −5.03500453792351870569526833067, −3.95293785852441471773103673868, −2.53346443654088687204795177824, 0.16512813895562496184733893796, 1.83722104063949320393331785019, 3.58455690220940094995600309875, 4.29945818549470963646631693054, 5.35723095008537165003048246418, 6.42112073282164452205862354831, 6.88724817052114236118213808031, 8.573579127314718795551220413209, 9.728470200430112308249421073272, 10.36199920321803873720785445134

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