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Properties

Label 2-600-120.53-c1-0-21
Degree $2$
Conductor $600$
Sign $0.989 - 0.146i$
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  + (−0.178 − 1.40i)2-s + (1.22 + 1.22i)3-s + (−1.93 + 0.5i)4-s + (1.5 − 1.93i)6-s + (1.04 + 2.62i)8-s + 2.99i·9-s + (−2.98 − 1.75i)12-s + (3.50 − 1.93i)16-s + (3.16 + 3.16i)17-s + (4.20 − 0.534i)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 + (−3.34 − 4.56i)32-s + ⋯
L(s)  = 1  + (−0.126 − 0.992i)2-s + (0.707 + 0.707i)3-s + (−0.968 + 0.250i)4-s + (0.612 − 0.790i)6-s + (0.370 + 0.929i)8-s + 0.999i·9-s + (−0.861 − 0.507i)12-s + (0.875 − 0.484i)16-s + (0.766 + 0.766i)17-s + (0.992 − 0.126i)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 + (−0.590 − 0.807i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.146i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 - 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.989 - 0.146i$
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: \((2,\ 600,\ (\ :1/2),\ 0.989 - 0.146i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.57495 + 0.116123i\)
\(L(\frac12)\) \(\approx\) \(1.57495 + 0.116123i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.178 + 1.40i)T \)
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

−10.52021834706540676490319759540, −9.746115400330502419834375424707, −9.364554416345818300690628142037, −8.103419179079603665722425711285, −7.72265151704389372858033512472, −5.79046609943871724601229970097, −4.81255747539468781289116418944, −3.72581916133783038393076671617, −3.00931711816584277496073710432, −1.59369139527753715641650658359, 0.975840896337774975061897820613, 2.82612920610447688176405419499, 4.07829139647210299920797243444, 5.35551014574230621852627095505, 6.32119859443720522936751010119, 7.22783086336465606568486161079, 7.88457195676260883855941981700, 8.638370252418924958232644440749, 9.582505386322565417585731851407, 10.18900890712814038333561374956

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