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Properties

Label 2-600-24.11-c1-0-46
Degree $2$
Conductor $600$
Sign $-0.0917 + 0.995i$
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  − 1.41i·2-s + (1.72 + 0.158i)3-s − 2.00·4-s + (0.224 − 2.43i)6-s + 2.82i·8-s + (2.94 + 0.548i)9-s − 3.78i·11-s + (−3.44 − 0.317i)12-s + 4.00·16-s − 8.02i·17-s + (0.775 − 4.17i)18-s + 6.34·19-s − 5.34·22-s + (−0.449 + 4.87i)24-s + (4.99 + 1.41i)27-s + ⋯
L(s)  = 1  − 0.999i·2-s + (0.995 + 0.0917i)3-s − 1.00·4-s + (0.0917 − 0.995i)6-s + 1.00i·8-s + (0.983 + 0.182i)9-s − 1.14i·11-s + (−0.995 − 0.0917i)12-s + 1.00·16-s − 1.94i·17-s + (0.182 − 0.983i)18-s + 1.45·19-s − 1.14·22-s + (−0.0917 + 0.995i)24-s + (0.962 + 0.272i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0917 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0917 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $-0.0917 + 0.995i$
Analytic conductor: \(4.79102\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ -0.0917 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27468 - 1.39753i\)
\(L(\frac12)\) \(\approx\) \(1.27468 - 1.39753i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
3 \( 1 + (-1.72 - 0.158i)T \)
5 \( 1 \)
good7 \( 1 - 7T^{2} \)
11 \( 1 + 3.78iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 8.02iT - 17T^{2} \)
19 \( 1 - 6.34T + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 10.8iT - 41T^{2} \)
43 \( 1 + 10T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 14.1iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 0.348T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 15.6T + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 - 17.0iT - 83T^{2} \)
89 \( 1 + 18.4iT - 89T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−10.29038686782357120573073187294, −9.559497205333416239037084077668, −8.929257845741613092907406687444, −8.063192221157579709872570939354, −7.15353935003074134333905240113, −5.52128835741893852585331153609, −4.54914426125641948665039221922, −3.28008792738676122344106674428, −2.76222378724683294052043134413, −1.10660997444743774770840227069, 1.70497655933536255691644910678, 3.44406971042512228721925907572, 4.31900200431173120586697709808, 5.45352945312898964327907056488, 6.66403648872221264584960976180, 7.41489738662164645969653340826, 8.155663064790663664371614167655, 8.953843010887635647387774603060, 9.804585939317584066777274715425, 10.42023729587679049268996051177

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