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Properties

Label 2-600-24.11-c1-0-39
Degree $2$
Conductor $600$
Sign $0.948 - 0.317i$
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  + (1.30 − 0.541i)2-s + (1.13 + 1.30i)3-s + (1.41 − 1.41i)4-s + (2.19 + 1.09i)6-s + 2.27i·7-s + (1.08 − 2.61i)8-s + (−0.414 + 2.97i)9-s + 4.20i·11-s + (3.45 + 0.239i)12-s − 3.21i·13-s + (1.23 + 2.97i)14-s − 4i·16-s − 1.53i·17-s + (1.06 + 4.10i)18-s + 4.82·19-s + ⋯
L(s)  = 1  + (0.923 − 0.382i)2-s + (0.656 + 0.754i)3-s + (0.707 − 0.707i)4-s + (0.895 + 0.445i)6-s + 0.859i·7-s + (0.382 − 0.923i)8-s + (−0.138 + 0.990i)9-s + 1.26i·11-s + (0.997 + 0.0692i)12-s − 0.891i·13-s + (0.328 + 0.794i)14-s − i·16-s − 0.371i·17-s + (0.251 + 0.967i)18-s + 1.10·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 - 0.317i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 - 0.317i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.948 - 0.317i$
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: \((2,\ 600,\ (\ :1/2),\ 0.948 - 0.317i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.07598 + 0.501833i\)
\(L(\frac12)\) \(\approx\) \(3.07598 + 0.501833i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.30 + 0.541i)T \)
3 \( 1 + (-1.13 - 1.30i)T \)
5 \( 1 \)
good7 \( 1 - 2.27iT - 7T^{2} \)
11 \( 1 - 4.20iT - 11T^{2} \)
13 \( 1 + 3.21iT - 13T^{2} \)
17 \( 1 + 1.53iT - 17T^{2} \)
19 \( 1 - 4.82T + 19T^{2} \)
23 \( 1 + 1.08T + 23T^{2} \)
29 \( 1 + 1.74T + 29T^{2} \)
31 \( 1 + 6.82iT - 31T^{2} \)
37 \( 1 + 7.76iT - 37T^{2} \)
41 \( 1 + 2.46iT - 41T^{2} \)
43 \( 1 + 8.70T + 43T^{2} \)
47 \( 1 + 1.08T + 47T^{2} \)
53 \( 1 + 11.0T + 53T^{2} \)
59 \( 1 - 4.20iT - 59T^{2} \)
61 \( 1 - 8.48iT - 61T^{2} \)
67 \( 1 - 2.27T + 67T^{2} \)
71 \( 1 - 11.8T + 71T^{2} \)
73 \( 1 - 4.54T + 73T^{2} \)
79 \( 1 - 0.485iT - 79T^{2} \)
83 \( 1 - 6.94iT - 83T^{2} \)
89 \( 1 + 8.40iT - 89T^{2} \)
97 \( 1 + 10.9T + 97T^{2} \)
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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

−10.72676956366595999461180743830, −9.775186081123877090030453250406, −9.389834644323846236518229424835, −8.014454960297427889402851949904, −7.15729547735051059935604286506, −5.68921459465208873454998284527, −5.08400013928632307558753449139, −4.04223442667285125102937827632, −2.96441550796902550393936071887, −2.06002322414140812073424266669, 1.50535337697377658415259683086, 3.09873978152497329976791981889, 3.75531416821443288493023808358, 5.07294154124326098870187002335, 6.34110835584661239681507460599, 6.86035259638470448698286353560, 7.87924782356689993641247478941, 8.468743076715037066512060741806, 9.636877767918537206717060375339, 10.95597335143084110235909973168

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