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Properties

Label 2-600-25.16-c1-0-10
Degree $2$
Conductor $600$
Sign $0.323 + 0.946i$
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  + (−0.309 − 0.951i)3-s + (−2.23 + 0.157i)5-s + 3.52·7-s + (−0.809 + 0.587i)9-s + (−0.246 − 0.179i)11-s + (2.26 − 1.64i)13-s + (0.839 + 2.07i)15-s + (1.20 − 3.72i)17-s + (0.119 − 0.368i)19-s + (−1.08 − 3.35i)21-s + (−2.19 − 1.59i)23-s + (4.95 − 0.704i)25-s + (0.809 + 0.587i)27-s + (−1.52 − 4.70i)29-s + (1.61 − 4.95i)31-s + ⋯
L(s)  = 1  + (−0.178 − 0.549i)3-s + (−0.997 + 0.0706i)5-s + 1.33·7-s + (−0.269 + 0.195i)9-s + (−0.0744 − 0.0540i)11-s + (0.628 − 0.456i)13-s + (0.216 + 0.535i)15-s + (0.293 − 0.902i)17-s + (0.0274 − 0.0844i)19-s + (−0.237 − 0.731i)21-s + (−0.457 − 0.332i)23-s + (0.990 − 0.140i)25-s + (0.155 + 0.113i)27-s + (−0.283 − 0.872i)29-s + (0.289 − 0.890i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.323 + 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.323 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.323 + 0.946i$
Analytic conductor: \(4.79102\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ 0.323 + 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03611 - 0.740428i\)
\(L(\frac12)\) \(\approx\) \(1.03611 - 0.740428i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 + (2.23 - 0.157i)T \)
good7 \( 1 - 3.52T + 7T^{2} \)
11 \( 1 + (0.246 + 0.179i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-2.26 + 1.64i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.20 + 3.72i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-0.119 + 0.368i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (2.19 + 1.59i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (1.52 + 4.70i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-1.61 + 4.95i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-6.69 + 4.86i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-6.63 + 4.82i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 11.3T + 43T^{2} \)
47 \( 1 + (-2.21 - 6.82i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (0.339 + 1.04i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-7.20 + 5.23i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (7.79 + 5.66i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (2.02 - 6.23i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (-0.749 - 2.30i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-11.0 - 8.05i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-1.06 - 3.29i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (4.49 - 13.8i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (-0.895 - 0.650i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-5.40 - 16.6i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.93937483452786399794915058803, −9.590205033320386447956563246952, −8.287800969765680302571892159111, −7.970464376300154139429131886999, −7.14022302803911554242553676243, −5.91785726844661632159059183196, −4.88946663419526507744440425064, −3.90533100502105590137756746516, −2.44057478774908595485947539457, −0.828175147702927485295780957480, 1.48714408738553184136551685751, 3.36952794416100879512697295437, 4.33072756551133503391606384178, 5.05422083632929394758873780845, 6.25752342966242984090547902691, 7.50435660185150892872759744780, 8.270505712858004334498932159388, 8.866033664745083339465982444441, 10.17612213628363044782878053518, 10.95345457693712609767677445236

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