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Properties

Label 2-600-24.11-c1-0-8
Degree $2$
Conductor $600$
Sign $-0.908 + 0.418i$
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 + (0.724 + 1.57i)3-s − 2.00·4-s + (−2.22 + 1.02i)6-s − 2.82i·8-s + (−1.94 + 2.28i)9-s + 6.61i·11-s + (−1.44 − 3.14i)12-s + 4.00·16-s − 2.36i·17-s + (−3.22 − 2.75i)18-s − 8.34·19-s − 9.34·22-s + (4.44 − 2.04i)24-s + (−5.00 − 1.41i)27-s + ⋯
L(s)  = 1  + 0.999i·2-s + (0.418 + 0.908i)3-s − 1.00·4-s + (−0.908 + 0.418i)6-s − 1.00i·8-s + (−0.649 + 0.760i)9-s + 1.99i·11-s + (−0.418 − 0.908i)12-s + 1.00·16-s − 0.574i·17-s + (−0.760 − 0.649i)18-s − 1.91·19-s − 1.99·22-s + (0.908 − 0.418i)24-s + (−0.962 − 0.272i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.908 + 0.418i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.908 + 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $-0.908 + 0.418i$
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: \((2,\ 600,\ (\ :1/2),\ -0.908 + 0.418i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.239286 - 1.09126i\)
\(L(\frac12)\) \(\approx\) \(0.239286 - 1.09126i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.724 - 1.57i)T \)
5 \( 1 \)
good7 \( 1 - 7T^{2} \)
11 \( 1 - 6.61iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 2.36iT - 17T^{2} \)
19 \( 1 + 8.34T + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 0.460iT - 41T^{2} \)
43 \( 1 - 10T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 14.1iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 14.3T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 13.6T + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 - 14.1iT - 83T^{2} \)
89 \( 1 - 12.7iT - 89T^{2} \)
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

−10.77134036012400299217004883952, −10.02963075738866796094413003815, −9.332454010540412247104919146260, −8.582494916036354261680445385096, −7.60135815847901278460459604886, −6.81609947955012021188071711995, −5.60323674393187027476246252548, −4.54193481752522462504601853870, −4.11122225182346979569168121830, −2.37672987597234901674568978512, 0.58663908602624407808599996061, 2.04832283639156490265104378621, 3.13996614076996600979206298663, 4.07720853240452779954542107476, 5.68472129577490533157520714386, 6.39494562166137588659466204729, 7.88321817747722108510718144503, 8.581294609966556397984440467977, 9.051711805165616786593532932683, 10.45277826903736670892428080480

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