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Properties

Label 2-300-4.3-c4-0-18
Degree $2$
Conductor $300$
Sign $0.829 - 0.558i$
Analytic cond. $31.0109$
Root an. cond. $5.56875$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.87 − 3.53i)2-s + 5.19i·3-s + (−8.94 − 13.2i)4-s + (18.3 + 9.76i)6-s − 7.86i·7-s + (−63.6 + 6.63i)8-s − 27·9-s + 61.8i·11-s + (68.9 − 46.4i)12-s − 47.0·13-s + (−27.7 − 14.7i)14-s + (−96.1 + 237. i)16-s + 239.·17-s + (−50.7 + 95.3i)18-s + 18.5i·19-s + ⋯
L(s)  = 1  + (0.469 − 0.882i)2-s + 0.577i·3-s + (−0.558 − 0.829i)4-s + (0.509 + 0.271i)6-s − 0.160i·7-s + (−0.994 + 0.103i)8-s − 0.333·9-s + 0.511i·11-s + (0.478 − 0.322i)12-s − 0.278·13-s + (−0.141 − 0.0754i)14-s + (−0.375 + 0.926i)16-s + 0.829·17-s + (−0.156 + 0.294i)18-s + 0.0513i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.829 - 0.558i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.829 - 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.829 - 0.558i$
Analytic conductor: \(31.0109\)
Root analytic conductor: \(5.56875\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :2),\ 0.829 - 0.558i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.712653507\)
\(L(\frac12)\) \(\approx\) \(1.712653507\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.87 + 3.53i)T \)
3 \( 1 - 5.19iT \)
5 \( 1 \)
good7 \( 1 + 7.86iT - 2.40e3T^{2} \)
11 \( 1 - 61.8iT - 1.46e4T^{2} \)
13 \( 1 + 47.0T + 2.85e4T^{2} \)
17 \( 1 - 239.T + 8.35e4T^{2} \)
19 \( 1 - 18.5iT - 1.30e5T^{2} \)
23 \( 1 - 580. iT - 2.79e5T^{2} \)
29 \( 1 - 254.T + 7.07e5T^{2} \)
31 \( 1 - 1.36e3iT - 9.23e5T^{2} \)
37 \( 1 + 1.08e3T + 1.87e6T^{2} \)
41 \( 1 - 2.01e3T + 2.82e6T^{2} \)
43 \( 1 - 716. iT - 3.41e6T^{2} \)
47 \( 1 - 2.54e3iT - 4.87e6T^{2} \)
53 \( 1 + 1.95e3T + 7.89e6T^{2} \)
59 \( 1 - 5.12e3iT - 1.21e7T^{2} \)
61 \( 1 - 7.40e3T + 1.38e7T^{2} \)
67 \( 1 - 2.55e3iT - 2.01e7T^{2} \)
71 \( 1 + 2.34e3iT - 2.54e7T^{2} \)
73 \( 1 + 3.54e3T + 2.83e7T^{2} \)
79 \( 1 - 4.94e3iT - 3.89e7T^{2} \)
83 \( 1 + 8.82e3iT - 4.74e7T^{2} \)
89 \( 1 + 1.29e4T + 6.27e7T^{2} \)
97 \( 1 + 1.31e4T + 8.85e7T^{2} \)
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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

−11.18307860485701166853545407444, −10.26799862954061010245465920340, −9.661001329550476160455031529391, −8.676956894589641807052088652478, −7.29212013698229678263696448848, −5.83129333256347783223009037075, −4.92103803599334291475798604610, −3.88218724684711693803628640150, −2.80550777147845003006822022334, −1.28171137127376151293806269400, 0.48658822595615798341007137681, 2.54412726320017201026562043139, 3.85761818110171989895787773880, 5.21153301067166682969634333181, 6.09354190526437310806488716152, 7.04403736661859100706280476485, 7.989914825098321533610731246441, 8.741844741861766923392126371632, 9.897314740809484295969155762192, 11.29101670778607022629332408654

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