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Properties

Label 2-300-25.6-c1-0-1
Degree $2$
Conductor $300$
Sign $0.997 - 0.0749i$
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 + (−1.57 + 1.59i)5-s + 4.32·7-s + (0.309 − 0.951i)9-s + (0.180 + 0.555i)11-s + (0.298 − 0.918i)13-s + (−0.336 + 2.21i)15-s + (1.88 + 1.36i)17-s + (4.35 + 3.16i)19-s + (3.49 − 2.54i)21-s + (−0.419 − 1.29i)23-s + (−0.0610 − 4.99i)25-s + (−0.309 − 0.951i)27-s + (0.571 − 0.415i)29-s + (−6.86 − 4.98i)31-s + ⋯
L(s)  = 1  + (0.467 − 0.339i)3-s + (−0.702 + 0.711i)5-s + 1.63·7-s + (0.103 − 0.317i)9-s + (0.0544 + 0.167i)11-s + (0.0828 − 0.254i)13-s + (−0.0868 + 0.570i)15-s + (0.456 + 0.331i)17-s + (0.998 + 0.725i)19-s + (0.763 − 0.554i)21-s + (−0.0875 − 0.269i)23-s + (−0.0122 − 0.999i)25-s + (−0.0594 − 0.183i)27-s + (0.106 − 0.0770i)29-s + (−1.23 − 0.896i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.997 - 0.0749i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.997 - 0.0749i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56922 + 0.0589039i\)
\(L(\frac12)\) \(\approx\) \(1.56922 + 0.0589039i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.57 - 1.59i)T \)
good7 \( 1 - 4.32T + 7T^{2} \)
11 \( 1 + (-0.180 - 0.555i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-0.298 + 0.918i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.88 - 1.36i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-4.35 - 3.16i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (0.419 + 1.29i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-0.571 + 0.415i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (6.86 + 4.98i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (1.89 - 5.81i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-3.41 + 10.5i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 7.03T + 43T^{2} \)
47 \( 1 + (7.33 - 5.33i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (7.20 - 5.23i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.25 - 6.93i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (1.48 + 4.57i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-0.304 - 0.221i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (8.54 - 6.20i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.0659 - 0.202i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-5.68 + 4.12i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (13.0 + 9.46i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (4.33 + 13.3i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (12.7 - 9.25i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.69269502291455007787892374364, −10.98570956030212669391978243156, −9.987397645352122685232857752444, −8.593928600619606192985898591592, −7.82649401386720081233719966250, −7.29001068255349012286176787930, −5.79543176415519640722057051460, −4.46789721949115305135216121691, −3.26521246908136153526298154657, −1.69663220115879902833391500232, 1.52068892563759896701500530207, 3.41697897100048037267751350460, 4.68992610641936173285053949771, 5.27477633206128880551917841418, 7.24370626129737133750959258532, 8.033141890037615530822151529868, 8.746950512373144541859214616902, 9.678678085949047345145319905223, 11.13922857034619178826192780500, 11.48400174432384306687164945446

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