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Properties

Label 2-300-4.3-c2-0-25
Degree $2$
Conductor $300$
Sign $-0.620 + 0.784i$
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  + (−1.88 − 0.656i)2-s + 1.73i·3-s + (3.13 + 2.48i)4-s + (1.13 − 3.27i)6-s − 9.55i·7-s + (−4.29 − 6.74i)8-s − 2.99·9-s + 9.92i·11-s + (−4.29 + 5.43i)12-s − 7.55·13-s + (−6.27 + 18.0i)14-s + (3.68 + 15.5i)16-s − 17.1·17-s + (5.66 + 1.97i)18-s − 26.1i·19-s + ⋯
L(s)  = 1  + (−0.944 − 0.328i)2-s + 0.577i·3-s + (0.784 + 0.620i)4-s + (0.189 − 0.545i)6-s − 1.36i·7-s + (−0.537 − 0.843i)8-s − 0.333·9-s + 0.902i·11-s + (−0.358 + 0.452i)12-s − 0.581·13-s + (−0.448 + 1.28i)14-s + (0.230 + 0.973i)16-s − 1.01·17-s + (0.314 + 0.109i)18-s − 1.37i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.620 + 0.784i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.620 + 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.620 + 0.784i$
Analytic conductor: \(8.17440\)
Root analytic conductor: \(2.85909\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1),\ -0.620 + 0.784i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.207797 - 0.429258i\)
\(L(\frac12)\) \(\approx\) \(0.207797 - 0.429258i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.88 + 0.656i)T \)
3 \( 1 - 1.73iT \)
5 \( 1 \)
good7 \( 1 + 9.55iT - 49T^{2} \)
11 \( 1 - 9.92iT - 121T^{2} \)
13 \( 1 + 7.55T + 169T^{2} \)
17 \( 1 + 17.1T + 289T^{2} \)
19 \( 1 + 26.1iT - 361T^{2} \)
23 \( 1 + 1.67iT - 529T^{2} \)
29 \( 1 - 0.350T + 841T^{2} \)
31 \( 1 + 46.0iT - 961T^{2} \)
37 \( 1 + 22.6T + 1.36e3T^{2} \)
41 \( 1 + 77.2T + 1.68e3T^{2} \)
43 \( 1 + 41.7iT - 1.84e3T^{2} \)
47 \( 1 + 14.0iT - 2.20e3T^{2} \)
53 \( 1 - 22.6T + 2.80e3T^{2} \)
59 \( 1 + 94.7iT - 3.48e3T^{2} \)
61 \( 1 - 38T + 3.72e3T^{2} \)
67 \( 1 - 29.8iT - 4.48e3T^{2} \)
71 \( 1 - 7.19iT - 5.04e3T^{2} \)
73 \( 1 - 34.3T + 5.32e3T^{2} \)
79 \( 1 + 46.0iT - 6.24e3T^{2} \)
83 \( 1 - 24.1iT - 6.88e3T^{2} \)
89 \( 1 + 100.T + 7.92e3T^{2} \)
97 \( 1 + 131.T + 9.40e3T^{2} \)
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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

−10.96018310393876646462956501529, −10.16577895692111277779865468482, −9.535474993978305717557983383722, −8.497149791195841298286889317471, −7.29418121062184059719887867733, −6.75685691239396965785397080451, −4.80265910588590143078352346866, −3.78085566632352833159707844003, −2.23182389557541751073299093947, −0.29852556652790469363851623908, 1.71287557504209260346883154989, 2.93752835409112067014494226337, 5.29331146043923213967393318655, 6.11125356884304016042115949454, 7.05052622780103964456940202780, 8.362112495324022188063929049897, 8.674797135755125254267638197285, 9.809000596335805883322280803858, 10.88980205584520853523269673885, 11.84846841879249946625943480914

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