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Properties

Label 2-600-24.11-c1-0-28
Degree $2$
Conductor $600$
Sign $-0.758 - 0.652i$
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  + (0.541 + 1.30i)2-s + (1.64 + 0.541i)3-s + (−1.41 + 1.41i)4-s + (0.183 + 2.44i)6-s + 3.29i·7-s + (−2.61 − 1.08i)8-s + (2.41 + 1.78i)9-s − 2.51i·11-s + (−3.09 + 1.56i)12-s + 4.65i·13-s + (−4.29 + 1.78i)14-s − 4i·16-s − 3.69i·17-s + (−1.02 + 4.11i)18-s − 0.828·19-s + ⋯
L(s)  = 1  + (0.382 + 0.923i)2-s + (0.949 + 0.312i)3-s + (−0.707 + 0.707i)4-s + (0.0748 + 0.997i)6-s + 1.24i·7-s + (−0.923 − 0.382i)8-s + (0.804 + 0.593i)9-s − 0.759i·11-s + (−0.892 + 0.450i)12-s + 1.29i·13-s + (−1.14 + 0.475i)14-s − i·16-s − 0.896i·17-s + (−0.240 + 0.970i)18-s − 0.190·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.758 - 0.652i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.758 - 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $-0.758 - 0.652i$
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: \((2,\ 600,\ (\ :1/2),\ -0.758 - 0.652i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.754607 + 2.03410i\)
\(L(\frac12)\) \(\approx\) \(0.754607 + 2.03410i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.541 - 1.30i)T \)
3 \( 1 + (-1.64 - 0.541i)T \)
5 \( 1 \)
good7 \( 1 - 3.29iT - 7T^{2} \)
11 \( 1 + 2.51iT - 11T^{2} \)
13 \( 1 - 4.65iT - 13T^{2} \)
17 \( 1 + 3.69iT - 17T^{2} \)
19 \( 1 + 0.828T + 19T^{2} \)
23 \( 1 - 2.61T + 23T^{2} \)
29 \( 1 + 6.08T + 29T^{2} \)
31 \( 1 + 1.17iT - 31T^{2} \)
37 \( 1 + 1.92iT - 37T^{2} \)
41 \( 1 - 8.59iT - 41T^{2} \)
43 \( 1 - 6.01T + 43T^{2} \)
47 \( 1 - 2.61T + 47T^{2} \)
53 \( 1 - 4.59T + 53T^{2} \)
59 \( 1 + 2.51iT - 59T^{2} \)
61 \( 1 + 8.48iT - 61T^{2} \)
67 \( 1 - 3.29T + 67T^{2} \)
71 \( 1 - 7.12T + 71T^{2} \)
73 \( 1 - 6.58T + 73T^{2} \)
79 \( 1 + 16.4iT - 79T^{2} \)
83 \( 1 + 9.37iT - 83T^{2} \)
89 \( 1 - 5.03iT - 89T^{2} \)
97 \( 1 - 2.72T + 97T^{2} \)
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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

−11.12375333547226011316335746327, −9.464676518196740746498310842017, −9.184453005182111534074818015131, −8.426488598494310525232657362827, −7.51201861861983771223905561065, −6.52940806539742973224567758007, −5.49134278047183224310513139542, −4.54706130557441535063299532371, −3.43637232884353396787640307647, −2.36254722579250332276960376991, 1.04439724721712428308399753087, 2.35403516505530799962645786220, 3.58903623298257969266990918330, 4.20367205536860403740816952178, 5.51750345893885566214650970516, 6.90203553847385330765665350653, 7.74651450039336078059537631284, 8.684339336975721207432010063306, 9.683260617559581196498445239362, 10.38114810810076112309597426281

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