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Properties

Label 2-300-25.16-c1-0-3
Degree $2$
Conductor $300$
Sign $-0.990 + 0.135i$
Analytic cond. $2.39551$
Root an. cond. $1.54774$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.309 − 0.951i)3-s + (−1.96 − 1.06i)5-s − 1.74·7-s + (−0.809 + 0.587i)9-s + (−1.87 − 1.36i)11-s + (−3.99 + 2.90i)13-s + (−0.407 + 2.19i)15-s + (1.15 − 3.55i)17-s + (−0.523 + 1.61i)19-s + (0.537 + 1.65i)21-s + (−7.02 − 5.10i)23-s + (2.72 + 4.19i)25-s + (0.809 + 0.587i)27-s + (0.964 + 2.96i)29-s + (2.95 − 9.09i)31-s + ⋯
L(s)  = 1  + (−0.178 − 0.549i)3-s + (−0.878 − 0.477i)5-s − 0.657·7-s + (−0.269 + 0.195i)9-s + (−0.565 − 0.411i)11-s + (−1.10 + 0.805i)13-s + (−0.105 + 0.567i)15-s + (0.280 − 0.862i)17-s + (−0.120 + 0.369i)19-s + (0.117 + 0.361i)21-s + (−1.46 − 1.06i)23-s + (0.544 + 0.838i)25-s + (0.155 + 0.113i)27-s + (0.179 + 0.551i)29-s + (0.530 − 1.63i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.135i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.135i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.990 + 0.135i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ -0.990 + 0.135i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0240342 - 0.353723i\)
\(L(\frac12)\) \(\approx\) \(0.0240342 - 0.353723i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 + (1.96 + 1.06i)T \)
good7 \( 1 + 1.74T + 7T^{2} \)
11 \( 1 + (1.87 + 1.36i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (3.99 - 2.90i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.15 + 3.55i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (0.523 - 1.61i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (7.02 + 5.10i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-0.964 - 2.96i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-2.95 + 9.09i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (4.34 - 3.15i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-7.05 + 5.12i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 2.86T + 43T^{2} \)
47 \( 1 + (2.61 + 8.04i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (0.415 + 1.27i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (3.54 - 2.57i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-12.4 - 9.01i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-2.85 + 8.77i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (-0.00728 - 0.0224i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-0.827 - 0.601i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (0.246 + 0.759i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-0.732 + 2.25i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (-3.93 - 2.85i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (1.06 + 3.29i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.65266321902540839412356431521, −10.37192419377727630579668248665, −9.377507137038105562070200644057, −8.253254708707529574507832465129, −7.47153097747855061326452286686, −6.48239131029764526719600456074, −5.21430579520330894216592877292, −4.02781204414478963335257559368, −2.49924201694392635208321723206, −0.25077322556532707292336558227, 2.79268241098197412373312381121, 3.87789613103136790405302001379, 5.06965294413434562002964270325, 6.29829094039378119788463429900, 7.48112775244180707069518033604, 8.226112987733870430657098434504, 9.701330579381335190511372448974, 10.24685205528830097791428801148, 11.15742599612600576979893187460, 12.25161118791317148911882143323

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