Properties

Label 2-300-20.19-c2-0-19
Degree $2$
Conductor $300$
Sign $0.922 - 0.385i$
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  + (−0.696 + 1.87i)2-s + 1.73·3-s + (−3.02 − 2.61i)4-s + (−1.20 + 3.24i)6-s + 5.46·7-s + (7.00 − 3.86i)8-s + 2.99·9-s − 11.0i·11-s + (−5.24 − 4.52i)12-s − 10.1i·13-s + (−3.80 + 10.2i)14-s + (2.35 + 15.8i)16-s − 24.4i·17-s + (−2.08 + 5.62i)18-s + 23.7i·19-s + ⋯
L(s)  = 1  + (−0.348 + 0.937i)2-s + 0.577·3-s + (−0.757 − 0.652i)4-s + (−0.201 + 0.541i)6-s + 0.781·7-s + (0.875 − 0.482i)8-s + 0.333·9-s − 1.00i·11-s + (−0.437 − 0.376i)12-s − 0.778i·13-s + (−0.272 + 0.732i)14-s + (0.147 + 0.989i)16-s − 1.43i·17-s + (−0.116 + 0.312i)18-s + 1.25i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 - 0.385i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.922 - 0.385i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.922 - 0.385i$
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: \((2,\ 300,\ (\ :1),\ 0.922 - 0.385i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.65798 + 0.332345i\)
\(L(\frac12)\) \(\approx\) \(1.65798 + 0.332345i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.696 - 1.87i)T \)
3 \( 1 - 1.73T \)
5 \( 1 \)
good7 \( 1 - 5.46T + 49T^{2} \)
11 \( 1 + 11.0iT - 121T^{2} \)
13 \( 1 + 10.1iT - 169T^{2} \)
17 \( 1 + 24.4iT - 289T^{2} \)
19 \( 1 - 23.7iT - 361T^{2} \)
23 \( 1 - 37.2T + 529T^{2} \)
29 \( 1 - 25.7T + 841T^{2} \)
31 \( 1 - 4.83iT - 961T^{2} \)
37 \( 1 - 35.6iT - 1.36e3T^{2} \)
41 \( 1 + 9.30T + 1.68e3T^{2} \)
43 \( 1 - 70.0T + 1.84e3T^{2} \)
47 \( 1 + 38.0T + 2.20e3T^{2} \)
53 \( 1 + 55.7iT - 2.80e3T^{2} \)
59 \( 1 - 55.5iT - 3.48e3T^{2} \)
61 \( 1 + 82.2T + 3.72e3T^{2} \)
67 \( 1 - 104.T + 4.48e3T^{2} \)
71 \( 1 + 76.7iT - 5.04e3T^{2} \)
73 \( 1 - 93.5iT - 5.32e3T^{2} \)
79 \( 1 + 49.3iT - 6.24e3T^{2} \)
83 \( 1 + 72.3T + 6.88e3T^{2} \)
89 \( 1 + 115.T + 7.92e3T^{2} \)
97 \( 1 + 72.9iT - 9.40e3T^{2} \)
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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

−11.39609585906949325566315390786, −10.44480400269391577182265185276, −9.436720559137421707648232867022, −8.481005671899029645519069998345, −7.909780303560047506611601690875, −6.89143133025687589473870218993, −5.61353624405605056977784211690, −4.70110401144361069861165762514, −3.12595118153884814549154701168, −1.05065975376765205110974217358, 1.47369844182490217849685811165, 2.60491794142069928976227159027, 4.11220712353385654350243541623, 4.90523938843947958389969848827, 6.89539193118649743491195819003, 7.889068674516725810980191522625, 8.843593021107013138395220236969, 9.488145224776577259080695771602, 10.65943089057005380680926880256, 11.26935537706672660710390694580

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