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Properties

Label 2-600-120.53-c1-0-12
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  + (−0.366 + 1.36i)2-s + 1.73·3-s + (−1.73 − i)4-s + (−0.633 + 2.36i)6-s + (−3 + 3i)7-s + (2 − 1.99i)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 + (−1.09 + 4.09i)18-s − 3.46·19-s + ⋯
L(s)  = 1  + (−0.258 + 0.965i)2-s + 1.00·3-s + (−0.866 − 0.5i)4-s + (−0.258 + 0.965i)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 + (−0.258 + 0.965i)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

\(L(1)\) \(\approx\) \(0.120886 + 1.03823i\)
\(L(\frac12)\) \(\approx\) \(0.120886 + 1.03823i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.366 - 1.36i)T \)
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

−10.60143219954242775940248667577, −9.784585282190626904675752792996, −9.203508902059139750079527239259, −8.454262736528866323896258483596, −7.61288888179146179030879092366, −6.70523878119252857532919466581, −5.78902645454577754701600107685, −4.68646116774650051939811359597, −3.39064757220482235426006404411, −2.16123962056558296224675194295, 0.54382896678450020750003076293, 2.47652265838920694019393402366, 3.21333558706539314563763654643, 4.14993825766593179578461007132, 5.36197801083135047673768917655, 7.27009831619509443943575100937, 7.58100770940570661728466851648, 8.675333828974192100161722282661, 9.743117271986654576266911804911, 10.10273664886732338670259145753

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