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Properties

Label 2-600-3.2-c2-0-24
Degree $2$
Conductor $600$
Sign $0.830 + 0.557i$
Analytic cond. $16.3488$
Root an. cond. $4.04336$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.67 − 2.49i)3-s + 12.7·7-s + (−3.41 − 8.32i)9-s + 12.6i·11-s + 7.44·13-s + 14.0i·17-s + 31.0·19-s + (21.3 − 31.8i)21-s + 7.50i·23-s + (−26.4 − 5.40i)27-s − 15.7i·29-s − 20.4·31-s + (31.4 + 21.1i)33-s − 12.9·37-s + (12.4 − 18.5i)39-s + ⋯
L(s)  = 1  + (0.557 − 0.830i)3-s + 1.82·7-s + (−0.379 − 0.925i)9-s + 1.14i·11-s + 0.572·13-s + 0.826i·17-s + 1.63·19-s + (1.01 − 1.51i)21-s + 0.326i·23-s + (−0.979 − 0.200i)27-s − 0.542i·29-s − 0.660·31-s + (0.953 + 0.639i)33-s − 0.349·37-s + (0.318 − 0.475i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.830 + 0.557i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.830 + 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.830 + 0.557i$
Analytic conductor: \(16.3488\)
Root analytic conductor: \(4.04336\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1),\ 0.830 + 0.557i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.766897284\)
\(L(\frac12)\) \(\approx\) \(2.766897284\)
\(L(2)\) 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 \( 1 + (-1.67 + 2.49i)T \)
5 \( 1 \)
good7 \( 1 - 12.7T + 49T^{2} \)
11 \( 1 - 12.6iT - 121T^{2} \)
13 \( 1 - 7.44T + 169T^{2} \)
17 \( 1 - 14.0iT - 289T^{2} \)
19 \( 1 - 31.0T + 361T^{2} \)
23 \( 1 - 7.50iT - 529T^{2} \)
29 \( 1 + 15.7iT - 841T^{2} \)
31 \( 1 + 20.4T + 961T^{2} \)
37 \( 1 + 12.9T + 1.36e3T^{2} \)
41 \( 1 + 13.8iT - 1.68e3T^{2} \)
43 \( 1 + 30.0T + 1.84e3T^{2} \)
47 \( 1 + 20.2iT - 2.20e3T^{2} \)
53 \( 1 - 29.1iT - 2.80e3T^{2} \)
59 \( 1 + 47.6iT - 3.48e3T^{2} \)
61 \( 1 - 43.0T + 3.72e3T^{2} \)
67 \( 1 - 0.630T + 4.48e3T^{2} \)
71 \( 1 + 90.4iT - 5.04e3T^{2} \)
73 \( 1 + 46.2T + 5.32e3T^{2} \)
79 \( 1 + 37.9T + 6.24e3T^{2} \)
83 \( 1 + 80.2iT - 6.88e3T^{2} \)
89 \( 1 + 140. iT - 7.92e3T^{2} \)
97 \( 1 + 10.3T + 9.40e3T^{2} \)
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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

−10.41396476311871700541601155915, −9.314134701726905695948348562907, −8.424565680649288899085947135620, −7.71690815219044333150634292152, −7.13918639757109544423255488208, −5.80056389038739185395261541862, −4.80659893878062076386318727476, −3.61253991154958823476286911086, −2.04636493696708919961635350024, −1.33226071022024788810921035178, 1.30525264062961639926406451337, 2.81811134768369245450279132256, 3.89205310578188981194410044861, 5.04030326790508833642211305033, 5.53929285526951954436739047358, 7.25685022578605240614809940284, 8.165728219613164475559739314562, 8.667035317307088630843344340669, 9.583237232988545261430407226862, 10.69518752467150542079070228535

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