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Properties

Label 2-600-120.53-c1-0-10
Degree $2$
Conductor $600$
Sign $0.573 - 0.819i$
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  + (−1.40 − 0.178i)2-s + (−1.22 − 1.22i)3-s + (1.93 + 0.5i)4-s + (1.5 + 1.93i)6-s + (−2.62 − 1.04i)8-s + 2.99i·9-s + (−1.75 − 2.98i)12-s + (3.50 + 1.93i)16-s + (3.16 + 3.16i)17-s + (0.534 − 4.20i)18-s − 7.74·19-s + (−6.32 + 6.32i)23-s + (1.93 + 4.5i)24-s + (3.67 − 3.67i)27-s + 8·31-s + (−4.56 − 3.34i)32-s + ⋯
L(s)  = 1  + (−0.992 − 0.126i)2-s + (−0.707 − 0.707i)3-s + (0.968 + 0.250i)4-s + (0.612 + 0.790i)6-s + (−0.929 − 0.370i)8-s + 0.999i·9-s + (−0.507 − 0.861i)12-s + (0.875 + 0.484i)16-s + (0.766 + 0.766i)17-s + (0.126 − 0.992i)18-s − 1.77·19-s + (−1.31 + 1.31i)23-s + (0.395 + 0.918i)24-s + (0.707 − 0.707i)27-s + 1.43·31-s + (−0.807 − 0.590i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.573 - 0.819i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.573 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.573 - 0.819i$
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: \((2,\ 600,\ (\ :1/2),\ 0.573 - 0.819i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.445221 + 0.231773i\)
\(L(\frac12)\) \(\approx\) \(0.445221 + 0.231773i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 + 0.178i)T \)
3 \( 1 + (1.22 + 1.22i)T \)
5 \( 1 \)
good7 \( 1 - 7iT^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 13iT^{2} \)
17 \( 1 + (-3.16 - 3.16i)T + 17iT^{2} \)
19 \( 1 + 7.74T + 19T^{2} \)
23 \( 1 + (6.32 - 6.32i)T - 23iT^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 + (-6.32 - 6.32i)T + 47iT^{2} \)
53 \( 1 + (-9.79 - 9.79i)T + 53iT^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 - 15.4iT - 61T^{2} \)
67 \( 1 + 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + 73iT^{2} \)
79 \( 1 + 16iT - 79T^{2} \)
83 \( 1 + (-2.44 - 2.44i)T + 83iT^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 97iT^{2} \)
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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

−10.56926471524268426730606511369, −10.23627542386327430374573514117, −8.956476471775064143046051419102, −8.041721897272840499587544350109, −7.45336160691741824514279072337, −6.29015487804159195851256571599, −5.82604898544274820759791807010, −4.14180346667588527106241603210, −2.48009406165604311021459714368, −1.30493500508135058350400568727, 0.44790707596659336550707019144, 2.37155183435243092267282651382, 3.90254728416299566739404515771, 5.12278252603997158164036051832, 6.20974384915864853270670932564, 6.80425564756937079336153133721, 8.149954225411931243398296451494, 8.787514120811993037415024072382, 9.992933219152730934189956027246, 10.21607758252545900693765644183

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