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Properties

Label 2-600-24.11-c1-0-65
Degree $2$
Conductor $600$
Sign $0.118 + 0.993i$
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  + (1.40 − 0.199i)2-s + (0.520 − 1.65i)3-s + (1.92 − 0.557i)4-s + (0.400 − 2.41i)6-s − 1.92i·7-s + (2.57 − 1.16i)8-s + (−2.45 − 1.72i)9-s + 4.02i·11-s + (0.0792 − 3.46i)12-s − 4.81i·13-s + (−0.383 − 2.69i)14-s + (3.37 − 2.14i)16-s + 5.23i·17-s + (−3.78 − 1.91i)18-s − 0.684·19-s + ⋯
L(s)  = 1  + (0.990 − 0.140i)2-s + (0.300 − 0.953i)3-s + (0.960 − 0.278i)4-s + (0.163 − 0.986i)6-s − 0.728i·7-s + (0.911 − 0.411i)8-s + (−0.819 − 0.573i)9-s + 1.21i·11-s + (0.0228 − 0.999i)12-s − 1.33i·13-s + (−0.102 − 0.721i)14-s + (0.844 − 0.535i)16-s + 1.26i·17-s + (−0.891 − 0.452i)18-s − 0.157·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.118 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.118 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.118 + 0.993i$
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: \((2,\ 600,\ (\ :1/2),\ 0.118 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.18783 - 1.94312i\)
\(L(\frac12)\) \(\approx\) \(2.18783 - 1.94312i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 + 0.199i)T \)
3 \( 1 + (-0.520 + 1.65i)T \)
5 \( 1 \)
good7 \( 1 + 1.92iT - 7T^{2} \)
11 \( 1 - 4.02iT - 11T^{2} \)
13 \( 1 + 4.81iT - 13T^{2} \)
17 \( 1 - 5.23iT - 17T^{2} \)
19 \( 1 + 0.684T + 19T^{2} \)
23 \( 1 + 1.72T + 23T^{2} \)
29 \( 1 - 6.99T + 29T^{2} \)
31 \( 1 + 4.23iT - 31T^{2} \)
37 \( 1 - 9.83iT - 37T^{2} \)
41 \( 1 - 3.44iT - 41T^{2} \)
43 \( 1 + 1.04T + 43T^{2} \)
47 \( 1 + 7.55T + 47T^{2} \)
53 \( 1 - 4.08T + 53T^{2} \)
59 \( 1 + 0.994iT - 59T^{2} \)
61 \( 1 - 3.16iT - 61T^{2} \)
67 \( 1 - 14.8T + 67T^{2} \)
71 \( 1 + 9.28T + 71T^{2} \)
73 \( 1 + 11.2T + 73T^{2} \)
79 \( 1 + 9.25iT - 79T^{2} \)
83 \( 1 - 7.15iT - 83T^{2} \)
89 \( 1 - 0.829iT - 89T^{2} \)
97 \( 1 - 1.45T + 97T^{2} \)
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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

−10.47011681239834513386292070736, −9.984042633168194835405343611089, −8.284368504781216376885508430684, −7.67150960426230306251013047733, −6.75302272048289866504562830634, −6.02541338116399677480620773075, −4.81040714835259061982445728844, −3.70336397103580035345157576727, −2.57365498917956308419530990063, −1.32225889162674717273138569481, 2.33886626628842526059723728210, 3.28719310343016083032386289689, 4.33003540876792570189455328890, 5.22902000849961253009020622285, 6.04787104788268031515726513921, 7.10768028383281617524175151019, 8.418949618699232275219283297419, 9.002411876680066694946910470465, 10.09020502832102981682992229263, 11.15653767000832188252493096206

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