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Properties

Label 2-600-120.59-c1-0-67
Degree $2$
Conductor $600$
Sign $-0.133 - 0.991i$
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  + (0.639 − 1.26i)2-s + (−0.730 − 1.57i)3-s + (−1.18 − 1.61i)4-s + (−2.44 − 0.0838i)6-s − 1.25·7-s + (−2.79 + 0.458i)8-s + (−1.93 + 2.29i)9-s + 3.02i·11-s + (−1.67 + 3.03i)12-s − 5.65·13-s + (−0.803 + 1.58i)14-s + (−1.20 + 3.81i)16-s + 2.45·17-s + (1.65 + 3.90i)18-s − 1.77·19-s + ⋯
L(s)  = 1  + (0.452 − 0.891i)2-s + (−0.421 − 0.906i)3-s + (−0.590 − 0.806i)4-s + (−0.999 − 0.0342i)6-s − 0.474·7-s + (−0.986 + 0.162i)8-s + (−0.644 + 0.764i)9-s + 0.911i·11-s + (−0.482 + 0.875i)12-s − 1.56·13-s + (−0.214 + 0.423i)14-s + (−0.301 + 0.953i)16-s + 0.595·17-s + (0.390 + 0.920i)18-s − 0.407·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.133 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.133 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $-0.133 - 0.991i$
Analytic conductor: \(4.79102\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ -0.133 - 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.221998 + 0.253983i\)
\(L(\frac12)\) \(\approx\) \(0.221998 + 0.253983i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.639 + 1.26i)T \)
3 \( 1 + (0.730 + 1.57i)T \)
5 \( 1 \)
good7 \( 1 + 1.25T + 7T^{2} \)
11 \( 1 - 3.02iT - 11T^{2} \)
13 \( 1 + 5.65T + 13T^{2} \)
17 \( 1 - 2.45T + 17T^{2} \)
19 \( 1 + 1.77T + 19T^{2} \)
23 \( 1 + 8.84iT - 23T^{2} \)
29 \( 1 + 3.79T + 29T^{2} \)
31 \( 1 + 5.19iT - 31T^{2} \)
37 \( 1 + 6.45T + 37T^{2} \)
41 \( 1 - 7.57iT - 41T^{2} \)
43 \( 1 + 4.37iT - 43T^{2} \)
47 \( 1 + 1.83iT - 47T^{2} \)
53 \( 1 - 12.0iT - 53T^{2} \)
59 \( 1 + 4.91iT - 59T^{2} \)
61 \( 1 + 8.16iT - 61T^{2} \)
67 \( 1 + 8.50iT - 67T^{2} \)
71 \( 1 - 7.00T + 71T^{2} \)
73 \( 1 - 4.59iT - 73T^{2} \)
79 \( 1 - 7.36iT - 79T^{2} \)
83 \( 1 + 15.7T + 83T^{2} \)
89 \( 1 + 3.65iT - 89T^{2} \)
97 \( 1 + 13.8iT - 97T^{2} \)
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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

−10.14600584571483161121652450071, −9.499567240225111446258604623479, −8.253597468165923249482818685964, −7.17124968703243147218044582220, −6.34367186790623759777925732774, −5.25184864866791859017004519062, −4.41812822374419527984670293843, −2.81846585406605608007952727243, −1.94020866316163966993502351659, −0.16043256876879550279721592287, 3.07813690800521293601262784884, 3.86270339858475823182966191412, 5.14899373326813608771229592088, 5.59303039791242470063091203549, 6.70594320125434465358543085382, 7.60998334904043865993986355923, 8.735155657267660993333160550277, 9.504991101875834520366098564954, 10.22169159964372700093255190201, 11.43720536908374606527025162476

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