Properties

Label 2-300-20.19-c2-0-32
Degree $2$
Conductor $300$
Sign $0.550 + 0.834i$
Analytic cond. $8.17440$
Root an. cond. $2.85909$
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  + (1.73 − i)2-s + 1.73·3-s + (1.99 − 3.46i)4-s + (2.99 − 1.73i)6-s + 6.92·7-s − 7.99i·8-s + 2.99·9-s + 6.92i·11-s + (3.46 − 5.99i)12-s − 2i·13-s + (11.9 − 6.92i)14-s + (−8 − 13.8i)16-s + 10i·17-s + (5.19 − 2.99i)18-s − 20.7i·19-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + 0.577·3-s + (0.499 − 0.866i)4-s + (0.499 − 0.288i)6-s + 0.989·7-s − 0.999i·8-s + 0.333·9-s + 0.629i·11-s + (0.288 − 0.499i)12-s − 0.153i·13-s + (0.857 − 0.494i)14-s + (−0.5 − 0.866i)16-s + 0.588i·17-s + (0.288 − 0.166i)18-s − 1.09i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.550 + 0.834i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.550 + 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.550 + 0.834i$
Analytic conductor: \(8.17440\)
Root analytic conductor: \(2.85909\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1),\ 0.550 + 0.834i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.04234 - 1.63693i\)
\(L(\frac12)\) \(\approx\) \(3.04234 - 1.63693i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.73 + i)T \)
3 \( 1 - 1.73T \)
5 \( 1 \)
good7 \( 1 - 6.92T + 49T^{2} \)
11 \( 1 - 6.92iT - 121T^{2} \)
13 \( 1 + 2iT - 169T^{2} \)
17 \( 1 - 10iT - 289T^{2} \)
19 \( 1 + 20.7iT - 361T^{2} \)
23 \( 1 + 27.7T + 529T^{2} \)
29 \( 1 - 26T + 841T^{2} \)
31 \( 1 - 6.92iT - 961T^{2} \)
37 \( 1 - 26iT - 1.36e3T^{2} \)
41 \( 1 - 58T + 1.68e3T^{2} \)
43 \( 1 + 48.4T + 1.84e3T^{2} \)
47 \( 1 + 69.2T + 2.20e3T^{2} \)
53 \( 1 - 74iT - 2.80e3T^{2} \)
59 \( 1 - 90.0iT - 3.48e3T^{2} \)
61 \( 1 - 26T + 3.72e3T^{2} \)
67 \( 1 + 6.92T + 4.48e3T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 - 46iT - 5.32e3T^{2} \)
79 \( 1 + 117. iT - 6.24e3T^{2} \)
83 \( 1 - 48.4T + 6.88e3T^{2} \)
89 \( 1 + 82T + 7.92e3T^{2} \)
97 \( 1 - 2iT - 9.40e3T^{2} \)
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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

−11.51471262739585528595735283617, −10.56106718862072075090429517554, −9.726396628451234073194241805030, −8.507123415659137338826331701552, −7.46814456218183493527974991868, −6.30656011217088864318490102768, −4.95709854892453157190780095567, −4.18673373290333875589633645639, −2.74253590983432627857121751044, −1.55079743004161313379367721290, 1.99775882615400387046723522220, 3.42802634189928113137753482572, 4.50967753413399769410181074881, 5.59487103176334962874095512518, 6.72355855467881294539626396495, 8.023619955106863678175656637872, 8.265643624782410565805602507192, 9.722858645894143494450655798880, 11.03336149648084367239279054602, 11.79307541212737285826606307415

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