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Properties

Label 2-600-15.14-c2-0-9
Degree $2$
Conductor $600$
Sign $0.532 - 0.846i$
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  + (−2.98 − 0.291i)3-s + 4.46i·7-s + (8.82 + 1.74i)9-s − 17.8i·11-s + 11.0i·13-s − 0.794·17-s − 26.5·19-s + (1.30 − 13.3i)21-s + 14.9·23-s + (−25.8 − 7.77i)27-s − 5.58i·29-s + 53.1·31-s + (−5.21 + 53.3i)33-s + 51.7i·37-s + (3.21 − 32.8i)39-s + ⋯
L(s)  = 1  + (−0.995 − 0.0972i)3-s + 0.637i·7-s + (0.981 + 0.193i)9-s − 1.62i·11-s + 0.846i·13-s − 0.0467·17-s − 1.39·19-s + (0.0619 − 0.634i)21-s + 0.649·23-s + (−0.957 − 0.287i)27-s − 0.192i·29-s + 1.71·31-s + (−0.157 + 1.61i)33-s + 1.39i·37-s + (0.0823 − 0.842i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.532 - 0.846i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.532 - 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.532 - 0.846i$
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: \((2,\ 600,\ (\ :1),\ 0.532 - 0.846i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.034951146\)
\(L(\frac12)\) \(\approx\) \(1.034951146\)
\(L(2)\) 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 \( 1 + (2.98 + 0.291i)T \)
5 \( 1 \)
good7 \( 1 - 4.46iT - 49T^{2} \)
11 \( 1 + 17.8iT - 121T^{2} \)
13 \( 1 - 11.0iT - 169T^{2} \)
17 \( 1 + 0.794T + 289T^{2} \)
19 \( 1 + 26.5T + 361T^{2} \)
23 \( 1 - 14.9T + 529T^{2} \)
29 \( 1 + 5.58iT - 841T^{2} \)
31 \( 1 - 53.1T + 961T^{2} \)
37 \( 1 - 51.7iT - 1.36e3T^{2} \)
41 \( 1 - 67.8iT - 1.68e3T^{2} \)
43 \( 1 - 40.8iT - 1.84e3T^{2} \)
47 \( 1 + 12.3T + 2.20e3T^{2} \)
53 \( 1 - 37.0T + 2.80e3T^{2} \)
59 \( 1 - 61.0iT - 3.48e3T^{2} \)
61 \( 1 - 97.8T + 3.72e3T^{2} \)
67 \( 1 - 3.02iT - 4.48e3T^{2} \)
71 \( 1 + 57.0iT - 5.04e3T^{2} \)
73 \( 1 - 31.4iT - 5.32e3T^{2} \)
79 \( 1 - 2.16T + 6.24e3T^{2} \)
83 \( 1 + 13.0T + 6.88e3T^{2} \)
89 \( 1 - 173. iT - 7.92e3T^{2} \)
97 \( 1 + 91.6iT - 9.40e3T^{2} \)
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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

−10.79199477030219513493017309465, −9.858679790908986407435758824724, −8.771157597783893611427949696888, −8.083185531445564269907112608672, −6.54665436216690620350203740223, −6.26703632246373621969170394420, −5.15049109098093997679379611485, −4.17655599858314012146397914551, −2.68292464023636526369885461157, −1.06448642645189007631369468104, 0.54209121309723807033523306879, 2.11661013971827147937698274996, 3.94959620310231771608413089505, 4.70732306745287551948862713884, 5.68381948866154318354506395614, 6.86104849899469436207734185238, 7.30534669913539579752460487330, 8.562635909186600985238606398057, 9.812937287834041373976740431725, 10.37914402578610101482525584795

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