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Properties

Label 2-600-15.14-c2-0-29
Degree $2$
Conductor $600$
Sign $0.358 + 0.933i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.358 + 0.933i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.358 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.358 + 0.933i$
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: \((2,\ 600,\ (\ :1),\ 0.358 + 0.933i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.369449260\)
\(L(\frac12)\) \(\approx\) \(2.369449260\)
\(L(2)\) 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

−10.39968876952203225750380473991, −9.338698714352858105047477229372, −8.301732781765625940151749222564, −8.035763782213823723599894375338, −6.76449137278482378232355069207, −5.79356978310498561856625315857, −4.37343924143225547733953574010, −3.51501270735580692257301563479, −2.47885798554829965229429362484, −0.792663440947294599779562564685, 1.79654529496826116142471373524, 2.55014642957317695774986832238, 4.05993697048470164823892859473, 4.76846887186571147589542847049, 6.36965677643798959948036357399, 7.08067713091623231004254822220, 8.146103051207359184281665083317, 8.816504227143859560690247882238, 9.736118662243766469662127073829, 10.27031482297329173194680252572

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