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Properties

Label 2-600-15.14-c2-0-15
Degree $2$
Conductor $600$
Sign $0.937 + 0.348i$
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  + (−0.323 − 2.98i)3-s + 4.72i·7-s + (−8.79 + 1.92i)9-s + 4.76i·11-s + 1.06i·13-s + 26.7·17-s + 8.12·19-s + (14.0 − 1.52i)21-s + 40.0·23-s + (8.59 + 25.5i)27-s + 20.8i·29-s − 33.7·31-s + (14.2 − 1.53i)33-s − 60.4i·37-s + (3.18 − 0.344i)39-s + ⋯
L(s)  = 1  + (−0.107 − 0.994i)3-s + 0.675i·7-s + (−0.976 + 0.214i)9-s + 0.433i·11-s + 0.0820i·13-s + 1.57·17-s + 0.427·19-s + (0.671 − 0.0727i)21-s + 1.74·23-s + (0.318 + 0.948i)27-s + 0.719i·29-s − 1.08·31-s + (0.430 − 0.0466i)33-s − 1.63i·37-s + (0.0815 − 0.00884i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.937 + 0.348i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.937 + 0.348i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.937 + 0.348i$
Analytic conductor: \(16.3488\)
Root analytic conductor: \(4.04336\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1),\ 0.937 + 0.348i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.726646899\)
\(L(\frac12)\) \(\approx\) \(1.726646899\)
\(L(2)\) 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 \( 1 + (0.323 + 2.98i)T \)
5 \( 1 \)
good7 \( 1 - 4.72iT - 49T^{2} \)
11 \( 1 - 4.76iT - 121T^{2} \)
13 \( 1 - 1.06iT - 169T^{2} \)
17 \( 1 - 26.7T + 289T^{2} \)
19 \( 1 - 8.12T + 361T^{2} \)
23 \( 1 - 40.0T + 529T^{2} \)
29 \( 1 - 20.8iT - 841T^{2} \)
31 \( 1 + 33.7T + 961T^{2} \)
37 \( 1 + 60.4iT - 1.36e3T^{2} \)
41 \( 1 + 59.2iT - 1.68e3T^{2} \)
43 \( 1 - 56.4iT - 1.84e3T^{2} \)
47 \( 1 + 9.68T + 2.20e3T^{2} \)
53 \( 1 - 93.1T + 2.80e3T^{2} \)
59 \( 1 - 17.4iT - 3.48e3T^{2} \)
61 \( 1 - 57.7T + 3.72e3T^{2} \)
67 \( 1 - 101. iT - 4.48e3T^{2} \)
71 \( 1 + 90.1iT - 5.04e3T^{2} \)
73 \( 1 + 40.0iT - 5.32e3T^{2} \)
79 \( 1 + 65.3T + 6.24e3T^{2} \)
83 \( 1 - 117.T + 6.88e3T^{2} \)
89 \( 1 - 119. iT - 7.92e3T^{2} \)
97 \( 1 + 15.2iT - 9.40e3T^{2} \)
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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

−10.54937099638758854554232232022, −9.320017655292645871148616152993, −8.681740292870689276513814596827, −7.50663119830798253948934380229, −7.06498334081977285086767362670, −5.73736184559116994020589538042, −5.23041013238452188459331210245, −3.46238970373857976356617675309, −2.33359979468183105383174281223, −1.03228038638120815547001808618, 0.888936910791857607510724065489, 3.01037909286585822049081043467, 3.78188903844127674154234430592, 4.97307901226986978504980981344, 5.69892507515903993820690888882, 6.94084782942687596975131703580, 7.957393964979554791633745540944, 8.886918275677927910225416263023, 9.819731937771384445858907865225, 10.38326506472089053653193378849

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