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Properties

Label 2-600-120.53-c1-0-45
Degree $2$
Conductor $600$
Sign $0.973 + 0.229i$
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  + (−1.36 + 0.366i)2-s + 1.73·3-s + (1.73 − i)4-s + (−2.36 + 0.633i)6-s + (3 − 3i)7-s + (−1.99 + 2i)8-s + 2.99·9-s + 3.46·11-s + (2.99 − 1.73i)12-s + (−3.46 + 3.46i)13-s + (−3 + 5.19i)14-s + (1.99 − 3.46i)16-s + (−4 − 4i)17-s + (−4.09 + 1.09i)18-s + 3.46·19-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)2-s + 1.00·3-s + (0.866 − 0.5i)4-s + (−0.965 + 0.258i)6-s + (1.13 − 1.13i)7-s + (−0.707 + 0.707i)8-s + 0.999·9-s + 1.04·11-s + (0.866 − 0.499i)12-s + (−0.960 + 0.960i)13-s + (−0.801 + 1.38i)14-s + (0.499 − 0.866i)16-s + (−0.970 − 0.970i)17-s + (−0.965 + 0.258i)18-s + 0.794·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.973 + 0.229i$
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: \((2,\ 600,\ (\ :1/2),\ 0.973 + 0.229i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55362 - 0.180894i\)
\(L(\frac12)\) \(\approx\) \(1.55362 - 0.180894i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 - 0.366i)T \)
3 \( 1 - 1.73T \)
5 \( 1 \)
good7 \( 1 + (-3 + 3i)T - 7iT^{2} \)
11 \( 1 - 3.46T + 11T^{2} \)
13 \( 1 + (3.46 - 3.46i)T - 13iT^{2} \)
17 \( 1 + (4 + 4i)T + 17iT^{2} \)
19 \( 1 - 3.46T + 19T^{2} \)
23 \( 1 + (1 - i)T - 23iT^{2} \)
29 \( 1 + 3.46iT - 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + 37iT^{2} \)
41 \( 1 - 6iT - 41T^{2} \)
43 \( 1 + (-1.73 + 1.73i)T - 43iT^{2} \)
47 \( 1 + (5 + 5i)T + 47iT^{2} \)
53 \( 1 + (-3.46 - 3.46i)T + 53iT^{2} \)
59 \( 1 - 10.3iT - 59T^{2} \)
61 \( 1 - 3.46iT - 61T^{2} \)
67 \( 1 + (-5.19 - 5.19i)T + 67iT^{2} \)
71 \( 1 + 12iT - 71T^{2} \)
73 \( 1 + (-6 - 6i)T + 73iT^{2} \)
79 \( 1 - 8iT - 79T^{2} \)
83 \( 1 + (1.73 + 1.73i)T + 83iT^{2} \)
89 \( 1 - 12T + 89T^{2} \)
97 \( 1 + (-6 + 6i)T - 97iT^{2} \)
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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

−10.36868265668616062857685211890, −9.492523317785964958865404481679, −9.008536584454729648163136411630, −7.936456383343181238317074444834, −7.27574760498958601833180942047, −6.73130174428924922456519240279, −4.92251577104727698457156705026, −3.97756103656526930628627780287, −2.34939265197782638473125188595, −1.27560408941363558251422972276, 1.62353533270321481062696354470, 2.47465057654521684936713360258, 3.70308866849059896852522422408, 5.11599260572765660389900422172, 6.49113432892990099912431898808, 7.57582031991896358192176824863, 8.241938010608025662111463107370, 8.937458309500283740540016283637, 9.535101441942163429878600955568, 10.57084115317543604475753398443

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