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Properties

Label 2-600-120.77-c1-0-30
Degree $2$
Conductor $600$
Sign $-0.229 - 0.973i$
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 + 1.73i·3-s + (1.73 + i)4-s + (−0.633 + 2.36i)6-s + (3 + 3i)7-s + (1.99 + 2i)8-s − 2.99·9-s − 3.46·11-s + (−1.73 + 2.99i)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 + 0.999i·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 + (−0.499 + 0.866i)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

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

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $-0.229 - 0.973i$
Analytic conductor: \(4.79102\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ -0.229 - 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69614 + 2.14317i\)
\(L(\frac12)\) \(\approx\) \(1.69614 + 2.14317i\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.73iT \)
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

−11.06732706728172480613326361191, −10.20087210577498040242262590972, −9.208442444630444979247641912579, −7.975098663010156309591212343800, −7.62035621458422665563381072200, −5.77464708711484932275919693149, −5.24166156883260021319969152706, −4.77362540568422118740579509908, −3.18838902252388561582168945069, −2.45980359560599632597324549402, 1.26066102682305683330088376154, 2.33396799141541103168253366832, 3.72084649044755309711271712456, 4.92130469306915122733391111024, 5.62756780386126041571200134989, 7.02507386940503344932230436137, 7.42167446866352496288689569485, 8.259042475786094186815982069883, 9.849678044004789039214857573879, 10.80720961831143247919439309986

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