Properties

Label 2-300-20.19-c2-0-26
Degree $2$
Conductor $300$
Sign $0.786 + 0.617i$
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  + (1.95 − 0.438i)2-s − 1.73·3-s + (3.61 − 1.71i)4-s + (−3.37 + 0.758i)6-s + 6.33·7-s + (6.30 − 4.92i)8-s + 2.99·9-s − 9.27i·11-s + (−6.26 + 2.96i)12-s + 18.5i·13-s + (12.3 − 2.77i)14-s + (10.1 − 12.3i)16-s − 13.9i·17-s + (5.85 − 1.31i)18-s − 17.2i·19-s + ⋯
L(s)  = 1  + (0.975 − 0.219i)2-s − 0.577·3-s + (0.904 − 0.427i)4-s + (−0.563 + 0.126i)6-s + 0.904·7-s + (0.788 − 0.615i)8-s + 0.333·9-s − 0.843i·11-s + (−0.521 + 0.246i)12-s + 1.42i·13-s + (0.882 − 0.198i)14-s + (0.634 − 0.772i)16-s − 0.818i·17-s + (0.325 − 0.0730i)18-s − 0.907i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.786 + 0.617i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.786 + 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.786 + 0.617i$
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: \((2,\ 300,\ (\ :1),\ 0.786 + 0.617i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.63373 - 0.910062i\)
\(L(\frac12)\) \(\approx\) \(2.63373 - 0.910062i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.95 + 0.438i)T \)
3 \( 1 + 1.73T \)
5 \( 1 \)
good7 \( 1 - 6.33T + 49T^{2} \)
11 \( 1 + 9.27iT - 121T^{2} \)
13 \( 1 - 18.5iT - 169T^{2} \)
17 \( 1 + 13.9iT - 289T^{2} \)
19 \( 1 + 17.2iT - 361T^{2} \)
23 \( 1 - 33.7T + 529T^{2} \)
29 \( 1 - 28.6T + 841T^{2} \)
31 \( 1 + 23.4iT - 961T^{2} \)
37 \( 1 - 67.3iT - 1.36e3T^{2} \)
41 \( 1 + 44.0T + 1.68e3T^{2} \)
43 \( 1 + 50.2T + 1.84e3T^{2} \)
47 \( 1 + 31.1T + 2.20e3T^{2} \)
53 \( 1 - 81.6iT - 2.80e3T^{2} \)
59 \( 1 - 19.2iT - 3.48e3T^{2} \)
61 \( 1 + 53.1T + 3.72e3T^{2} \)
67 \( 1 + 4.49T + 4.48e3T^{2} \)
71 \( 1 + 13.3iT - 5.04e3T^{2} \)
73 \( 1 - 40.8iT - 5.32e3T^{2} \)
79 \( 1 - 141. iT - 6.24e3T^{2} \)
83 \( 1 + 69.8T + 6.88e3T^{2} \)
89 \( 1 - 46.3T + 7.92e3T^{2} \)
97 \( 1 + 68.5iT - 9.40e3T^{2} \)
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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

−11.39860091778740639839723305787, −11.08555370191022826237542541081, −9.781321200597589011044435181106, −8.538532161946739431996191005786, −7.12165230944355317295240815709, −6.40867465493973650529472281385, −5.07019388305197108220474101385, −4.53706482596483509232735908245, −2.93408774777415631668212880250, −1.31360433120439707739987049445, 1.67941239512989344135799976278, 3.38309263270107467409582382074, 4.76703401560459060506325161398, 5.37930942214753905501100949535, 6.54479197693088193508791705789, 7.58695790953343761706611549127, 8.411526378020761849535536958837, 10.22936520602631444839395531049, 10.80047999117731932072218376382, 11.81424935300202219767251448277

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