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Properties

Label 2-600-120.53-c1-0-15
Degree $2$
Conductor $600$
Sign $0.850 - 0.525i$
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 − i)2-s + (−1.22 + 1.22i)3-s + 2i·4-s + 2.44·6-s + (2 − 2i)8-s − 2.99i·9-s + 4.89·11-s + (−2.44 − 2.44i)12-s + (−2.44 + 2.44i)13-s − 4·16-s + (−2 − 2i)17-s + (−2.99 + 2.99i)18-s + (−4.89 − 4.89i)22-s + (4 − 4i)23-s + 4.89i·24-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + (−0.707 + 0.707i)3-s + i·4-s + 0.999·6-s + (0.707 − 0.707i)8-s − 0.999i·9-s + 1.47·11-s + (−0.707 − 0.707i)12-s + (−0.679 + 0.679i)13-s − 16-s + (−0.485 − 0.485i)17-s + (−0.707 + 0.707i)18-s + (−1.04 − 1.04i)22-s + (0.834 − 0.834i)23-s + 0.999i·24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.850 - 0.525i$
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: \((2,\ 600,\ (\ :1/2),\ 0.850 - 0.525i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.755922 + 0.214741i\)
\(L(\frac12)\) \(\approx\) \(0.755922 + 0.214741i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + i)T \)
3 \( 1 + (1.22 - 1.22i)T \)
5 \( 1 \)
good7 \( 1 - 7iT^{2} \)
11 \( 1 - 4.89T + 11T^{2} \)
13 \( 1 + (2.44 - 2.44i)T - 13iT^{2} \)
17 \( 1 + (2 + 2i)T + 17iT^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + (-4 + 4i)T - 23iT^{2} \)
29 \( 1 - 9.79iT - 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 + (-7.34 - 7.34i)T + 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + (2.44 - 2.44i)T - 43iT^{2} \)
47 \( 1 + (-8 - 8i)T + 47iT^{2} \)
53 \( 1 + 53iT^{2} \)
59 \( 1 + 14.6iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + (-7.34 - 7.34i)T + 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + 73iT^{2} \)
79 \( 1 - 14iT - 79T^{2} \)
83 \( 1 + 83iT^{2} \)
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

−10.91618635373954567477358974659, −9.719190584963406969802881592178, −9.333750185190433010967437100123, −8.523475917963840847117389224545, −7.02050509063293784353739478071, −6.50231946059214428590577074740, −4.83825600340492917680013539899, −4.13905861879989977376199735507, −2.90627671820335470093898059106, −1.16185462899952953042860683359, 0.75265880307442498912615705373, 2.13317254319976829544608293468, 4.26434304899191110716484966358, 5.47320594146263700198060073408, 6.20982010820383205626515288501, 7.04743246135862329047633419135, 7.74223135049702964264244641390, 8.743339572621577243980977069259, 9.635981160997039780835213297031, 10.52267624712126921727177213776

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