Properties

Label 2-300-20.19-c2-0-29
Degree $2$
Conductor $300$
Sign $-0.717 + 0.696i$
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  + (−0.305 − 1.97i)2-s + 1.73·3-s + (−3.81 + 1.20i)4-s + (−0.529 − 3.42i)6-s + 0.329·7-s + (3.55 + 7.16i)8-s + 2.99·9-s − 20.4i·11-s + (−6.60 + 2.09i)12-s + 0.416i·13-s + (−0.100 − 0.652i)14-s + (13.0 − 9.21i)16-s − 18.5i·17-s + (−0.917 − 5.92i)18-s − 12.4i·19-s + ⋯
L(s)  = 1  + (−0.152 − 0.988i)2-s + 0.577·3-s + (−0.953 + 0.302i)4-s + (−0.0882 − 0.570i)6-s + 0.0471·7-s + (0.444 + 0.895i)8-s + 0.333·9-s − 1.86i·11-s + (−0.550 + 0.174i)12-s + 0.0320i·13-s + (−0.00720 − 0.0465i)14-s + (0.817 − 0.575i)16-s − 1.09i·17-s + (−0.0509 − 0.329i)18-s − 0.655i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.717 + 0.696i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.717 + 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.717 + 0.696i$
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: \((2,\ 300,\ (\ :1),\ -0.717 + 0.696i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.549169 - 1.35421i\)
\(L(\frac12)\) \(\approx\) \(0.549169 - 1.35421i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.305 + 1.97i)T \)
3 \( 1 - 1.73T \)
5 \( 1 \)
good7 \( 1 - 0.329T + 49T^{2} \)
11 \( 1 + 20.4iT - 121T^{2} \)
13 \( 1 - 0.416iT - 169T^{2} \)
17 \( 1 + 18.5iT - 289T^{2} \)
19 \( 1 + 12.4iT - 361T^{2} \)
23 \( 1 + 23.2T + 529T^{2} \)
29 \( 1 - 23.9T + 841T^{2} \)
31 \( 1 + 42.0iT - 961T^{2} \)
37 \( 1 - 50.9iT - 1.36e3T^{2} \)
41 \( 1 - 46.7T + 1.68e3T^{2} \)
43 \( 1 + 55.5T + 1.84e3T^{2} \)
47 \( 1 - 81.7T + 2.20e3T^{2} \)
53 \( 1 - 29.9iT - 2.80e3T^{2} \)
59 \( 1 + 24.3iT - 3.48e3T^{2} \)
61 \( 1 + 74.8T + 3.72e3T^{2} \)
67 \( 1 - 72.8T + 4.48e3T^{2} \)
71 \( 1 - 39.2iT - 5.04e3T^{2} \)
73 \( 1 - 46.5iT - 5.32e3T^{2} \)
79 \( 1 + 101. iT - 6.24e3T^{2} \)
83 \( 1 + 5.88T + 6.88e3T^{2} \)
89 \( 1 - 61.0T + 7.92e3T^{2} \)
97 \( 1 - 95.5iT - 9.40e3T^{2} \)
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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

−11.28025149074382150630749095225, −10.25022630881969290470855914935, −9.301142165431993674076999949180, −8.529280168056121421336207861560, −7.71320256682818825977828943426, −6.08569901344175527844984269442, −4.72774144955877384885740337525, −3.46935236054594079043459510400, −2.53944535675987756744664578954, −0.73344348095053977453551050497, 1.81974822531947962653226259124, 3.85644715821131735858740175444, 4.79595816117410546408955742829, 6.13106971521525050414096337724, 7.18263776984898799710724534705, 7.939703846057187224231264137684, 8.870464841469313988281714255951, 9.874265349641358540812912378547, 10.44990725197283026570739822664, 12.34246242005556468923097304438

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