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Properties

Label 2-600-8.5-c1-0-1
Degree $2$
Conductor $600$
Sign $-0.0864 - 0.996i$
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  + (0.671 − 1.24i)2-s + i·3-s + (−1.09 − 1.67i)4-s + (1.24 + 0.671i)6-s − 4.68·7-s + (−2.81 + 0.244i)8-s − 9-s + 2.29i·11-s + (1.67 − 1.09i)12-s + 4.97i·13-s + (−3.14 + 5.83i)14-s + (−1.58 + 3.67i)16-s + 2.97·17-s + (−0.671 + 1.24i)18-s + 2.68i·19-s + ⋯
L(s)  = 1  + (0.474 − 0.880i)2-s + 0.577i·3-s + (−0.549 − 0.835i)4-s + (0.508 + 0.274i)6-s − 1.77·7-s + (−0.996 + 0.0864i)8-s − 0.333·9-s + 0.691i·11-s + (0.482 − 0.317i)12-s + 1.38i·13-s + (−0.840 + 1.55i)14-s + (−0.396 + 0.917i)16-s + 0.722·17-s + (−0.158 + 0.293i)18-s + 0.616i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0864 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0864 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $-0.0864 - 0.996i$
Analytic conductor: \(4.79102\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ -0.0864 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.388482 + 0.423672i\)
\(L(\frac12)\) \(\approx\) \(0.388482 + 0.423672i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.671 + 1.24i)T \)
3 \( 1 - iT \)
5 \( 1 \)
good7 \( 1 + 4.68T + 7T^{2} \)
11 \( 1 - 2.29iT - 11T^{2} \)
13 \( 1 - 4.97iT - 13T^{2} \)
17 \( 1 - 2.97T + 17T^{2} \)
19 \( 1 - 2.68iT - 19T^{2} \)
23 \( 1 + 2.68T + 23T^{2} \)
29 \( 1 - 2iT - 29T^{2} \)
31 \( 1 + 6.97T + 31T^{2} \)
37 \( 1 + 4.39iT - 37T^{2} \)
41 \( 1 + 11.3T + 41T^{2} \)
43 \( 1 + 9.37iT - 43T^{2} \)
47 \( 1 + 7.27T + 47T^{2} \)
53 \( 1 - 2iT - 53T^{2} \)
59 \( 1 - 1.70iT - 59T^{2} \)
61 \( 1 - 4.58iT - 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 - 0.585T + 71T^{2} \)
73 \( 1 - 6T + 73T^{2} \)
79 \( 1 - 1.02T + 79T^{2} \)
83 \( 1 - 13.3iT - 83T^{2} \)
89 \( 1 - 3.37T + 89T^{2} \)
97 \( 1 - 3.95T + 97T^{2} \)
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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

−10.76322879949399764225587840662, −9.954973714189730196098248487901, −9.552638673523081912657956662769, −8.770020051111890444544062322693, −7.06956945417893704118656753246, −6.19788783365739776397584214908, −5.21351481611910029991471441419, −3.95118443388469199361381746126, −3.41806769612818077528683422137, −2.02236488104300770200098635607, 0.25918385832446682173167772046, 3.00927015937391533216004507006, 3.49924428836250819257477577853, 5.21696762738697426075711321086, 6.06189661499978422921035344586, 6.65121686884385132888141417916, 7.65002519310158020654506620795, 8.415130191167130876403941123962, 9.441751770946556789648070162931, 10.22943166113463301322877326614

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