Properties

Label 2-300-25.21-c1-0-3
Degree $2$
Conductor $300$
Sign $0.699 + 0.714i$
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  + (−0.809 − 0.587i)3-s + (2.04 − 0.909i)5-s + 0.747·7-s + (0.309 + 0.951i)9-s + (0.0646 − 0.198i)11-s + (−0.773 − 2.38i)13-s + (−2.18 − 0.464i)15-s + (5.51 − 4.00i)17-s + (−1.00 + 0.731i)19-s + (−0.604 − 0.439i)21-s + (1.00 − 3.09i)23-s + (3.34 − 3.71i)25-s + (0.309 − 0.951i)27-s + (4.19 + 3.04i)29-s + (−3.02 + 2.19i)31-s + ⋯
L(s)  = 1  + (−0.467 − 0.339i)3-s + (0.913 − 0.406i)5-s + 0.282·7-s + (0.103 + 0.317i)9-s + (0.0194 − 0.0599i)11-s + (−0.214 − 0.660i)13-s + (−0.564 − 0.120i)15-s + (1.33 − 0.972i)17-s + (−0.231 + 0.167i)19-s + (−0.131 − 0.0958i)21-s + (0.209 − 0.644i)23-s + (0.669 − 0.743i)25-s + (0.0594 − 0.183i)27-s + (0.778 + 0.565i)29-s + (−0.543 + 0.394i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.699 + 0.714i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.699 + 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.699 + 0.714i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.699 + 0.714i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22455 - 0.514754i\)
\(L(\frac12)\) \(\approx\) \(1.22455 - 0.514754i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.809 + 0.587i)T \)
5 \( 1 + (-2.04 + 0.909i)T \)
good7 \( 1 - 0.747T + 7T^{2} \)
11 \( 1 + (-0.0646 + 0.198i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (0.773 + 2.38i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-5.51 + 4.00i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.00 - 0.731i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-1.00 + 3.09i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-4.19 - 3.04i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (3.02 - 2.19i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.607 - 1.86i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-0.993 - 3.05i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 12.7T + 43T^{2} \)
47 \( 1 + (-5.24 - 3.81i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (3.35 + 2.43i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-3.61 - 11.1i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (3.85 - 11.8i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (2.35 - 1.71i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (5.29 + 3.85i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.778 - 2.39i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-8.28 - 6.02i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-4.59 + 3.33i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (0.284 - 0.876i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-12.5 - 9.13i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.79395889091470212474639251377, −10.55864153635753311223659151493, −9.917180258399137693532960178877, −8.782266349037468392374329405340, −7.75356742277304534176730700084, −6.61900207027993382267166696958, −5.55124303368304231717163739884, −4.83003744634610638714514718882, −2.87989761038925967401245941719, −1.23198642295838561295038337509, 1.79397027610451142920268268071, 3.48931671844367102238583722718, 4.92604053570603960784421293138, 5.88650912630169539830950299606, 6.78435835930095847033835667391, 8.029167243565795665860029948267, 9.315040673708826511051237012177, 10.02741851376170728570092127554, 10.83368620541611594888499315154, 11.76302094595952965535139037937

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