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Properties

Label 2-600-24.11-c1-0-60
Degree $2$
Conductor $600$
Sign $-0.477 + 0.878i$
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  + (−0.847 + 1.13i)2-s + (0.242 − 1.71i)3-s + (−0.562 − 1.91i)4-s + (1.73 + 1.72i)6-s − 3.08i·7-s + (2.64 + 0.990i)8-s + (−2.88 − 0.831i)9-s + 2.54i·11-s + (−3.42 + 0.499i)12-s − 5.06i·13-s + (3.49 + 2.61i)14-s + (−3.36 + 2.15i)16-s + 0.214i·17-s + (3.38 − 2.55i)18-s + 2.60·19-s + ⋯
L(s)  = 1  + (−0.599 + 0.800i)2-s + (0.139 − 0.990i)3-s + (−0.281 − 0.959i)4-s + (0.708 + 0.705i)6-s − 1.16i·7-s + (0.936 + 0.350i)8-s + (−0.960 − 0.277i)9-s + 0.767i·11-s + (−0.989 + 0.144i)12-s − 1.40i·13-s + (0.934 + 0.700i)14-s + (−0.841 + 0.539i)16-s + 0.0519i·17-s + (0.797 − 0.602i)18-s + 0.598·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.477 + 0.878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.477 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $-0.477 + 0.878i$
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: \((2,\ 600,\ (\ :1/2),\ -0.477 + 0.878i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.381731 - 0.642223i\)
\(L(\frac12)\) \(\approx\) \(0.381731 - 0.642223i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.847 - 1.13i)T \)
3 \( 1 + (-0.242 + 1.71i)T \)
5 \( 1 \)
good7 \( 1 + 3.08iT - 7T^{2} \)
11 \( 1 - 2.54iT - 11T^{2} \)
13 \( 1 + 5.06iT - 13T^{2} \)
17 \( 1 - 0.214iT - 17T^{2} \)
19 \( 1 - 2.60T + 19T^{2} \)
23 \( 1 + 4.47T + 23T^{2} \)
29 \( 1 + 7.86T + 29T^{2} \)
31 \( 1 - 4.58iT - 31T^{2} \)
37 \( 1 + 7.67iT - 37T^{2} \)
41 \( 1 + 9.26iT - 41T^{2} \)
43 \( 1 + 11.4T + 43T^{2} \)
47 \( 1 - 10.5T + 47T^{2} \)
53 \( 1 + 9.51T + 53T^{2} \)
59 \( 1 + 0.428iT - 59T^{2} \)
61 \( 1 - 1.11iT - 61T^{2} \)
67 \( 1 + 2.35T + 67T^{2} \)
71 \( 1 - 6.12T + 71T^{2} \)
73 \( 1 - 12.0T + 73T^{2} \)
79 \( 1 + 11.6iT - 79T^{2} \)
83 \( 1 - 2.29iT - 83T^{2} \)
89 \( 1 + 12.4iT - 89T^{2} \)
97 \( 1 - 8.04T + 97T^{2} \)
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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

−10.31208789885305155013075883443, −9.384255582963050070209948745846, −8.320879980208809366333755647826, −7.44072027137516399115479700351, −7.25224027758402710137618638410, −6.03902280412044471467110970751, −5.16540975725276166436219009069, −3.66422747818564992249981663383, −1.86698780667373821314737884337, −0.48960063557071366626937892985, 2.02766869069327549103996091332, 3.14181632568653571432203369754, 4.12359369061119914520308018695, 5.23453835946029626528873958866, 6.36279551626914987814499460572, 7.917737380008904449532024241537, 8.635111962187899902679860297981, 9.422030174472596840352424713053, 9.801351796807707038528799544391, 11.09234535592157680622881513976

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