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Properties

Label 2-600-40.29-c1-0-5
Degree $2$
Conductor $600$
Sign $-0.100 - 0.994i$
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  + (−1.38 + 0.264i)2-s + 3-s + (1.85 − 0.735i)4-s + (−1.38 + 0.264i)6-s + 0.941i·7-s + (−2.38 + 1.51i)8-s + 9-s + 4.49i·11-s + (1.85 − 0.735i)12-s − 5.55·13-s + (−0.249 − 1.30i)14-s + (2.91 − 2.73i)16-s + 7.55i·17-s + (−1.38 + 0.264i)18-s − 1.05i·19-s + ⋯
L(s)  = 1  + (−0.982 + 0.187i)2-s + 0.577·3-s + (0.929 − 0.367i)4-s + (−0.567 + 0.108i)6-s + 0.355i·7-s + (−0.844 + 0.535i)8-s + 0.333·9-s + 1.35i·11-s + (0.536 − 0.212i)12-s − 1.54·13-s + (−0.0665 − 0.349i)14-s + (0.729 − 0.683i)16-s + 1.83i·17-s + (−0.327 + 0.0623i)18-s − 0.242i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.100 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.100 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $-0.100 - 0.994i$
Analytic conductor: \(4.79102\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ -0.100 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.625520 + 0.692192i\)
\(L(\frac12)\) \(\approx\) \(0.625520 + 0.692192i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.38 - 0.264i)T \)
3 \( 1 - T \)
5 \( 1 \)
good7 \( 1 - 0.941iT - 7T^{2} \)
11 \( 1 - 4.49iT - 11T^{2} \)
13 \( 1 + 5.55T + 13T^{2} \)
17 \( 1 - 7.55iT - 17T^{2} \)
19 \( 1 + 1.05iT - 19T^{2} \)
23 \( 1 - 1.05iT - 23T^{2} \)
29 \( 1 - 2iT - 29T^{2} \)
31 \( 1 - 3.55T + 31T^{2} \)
37 \( 1 - 7.43T + 37T^{2} \)
41 \( 1 + 3.88T + 41T^{2} \)
43 \( 1 + 1.88T + 43T^{2} \)
47 \( 1 + 10.0iT - 47T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 - 8.49iT - 59T^{2} \)
61 \( 1 - 8.99iT - 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 + 12.9T + 71T^{2} \)
73 \( 1 - 6iT - 73T^{2} \)
79 \( 1 + 11.5T + 79T^{2} \)
83 \( 1 - 5.88T + 83T^{2} \)
89 \( 1 - 4.11T + 89T^{2} \)
97 \( 1 - 17.1iT - 97T^{2} \)
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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

−10.37172012975315399858094192926, −10.02837993468662169904016174427, −9.135232051800373131115585465901, −8.320144520133779320125028938056, −7.46158514654592566751409972413, −6.79829668498691708963294715896, −5.56166351968383264339307476619, −4.31585325601172174806390557034, −2.67846506628785444712207839419, −1.78370358698595718345923454787, 0.64593008690498888212488933530, 2.45652533239228763939897541453, 3.23804994450226999943665244299, 4.75519815235280460863989841102, 6.16487478886169171935078567876, 7.25717821213129927572938990673, 7.81300291707031309049807851783, 8.772280095166144514428854528927, 9.573083403889929673037364002791, 10.15293963070984125999916397544

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