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Properties

Label 2-300-75.53-c1-0-1
Degree $2$
Conductor $300$
Sign $0.465 - 0.884i$
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  + (−1.58 − 0.696i)3-s + (−1.99 + 1.00i)5-s + (0.814 − 0.814i)7-s + (2.03 + 2.20i)9-s + (3.46 + 1.12i)11-s + (−2.63 + 5.16i)13-s + (3.86 − 0.207i)15-s + (−0.902 − 0.142i)17-s + (4.29 + 5.91i)19-s + (−1.85 + 0.724i)21-s + (3.05 + 6.00i)23-s + (2.97 − 4.02i)25-s + (−1.68 − 4.91i)27-s + (−4.30 − 3.12i)29-s + (−1.78 + 1.29i)31-s + ⋯
L(s)  = 1  + (−0.915 − 0.401i)3-s + (−0.892 + 0.450i)5-s + (0.307 − 0.307i)7-s + (0.677 + 0.735i)9-s + (1.04 + 0.339i)11-s + (−0.730 + 1.43i)13-s + (0.998 − 0.0536i)15-s + (−0.218 − 0.0346i)17-s + (0.986 + 1.35i)19-s + (−0.405 + 0.158i)21-s + (0.637 + 1.25i)23-s + (0.594 − 0.804i)25-s + (−0.324 − 0.945i)27-s + (−0.799 − 0.580i)29-s + (−0.321 + 0.233i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.465 - 0.884i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.465 - 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.465 - 0.884i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.465 - 0.884i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.662280 + 0.399890i\)
\(L(\frac12)\) \(\approx\) \(0.662280 + 0.399890i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.58 + 0.696i)T \)
5 \( 1 + (1.99 - 1.00i)T \)
good7 \( 1 + (-0.814 + 0.814i)T - 7iT^{2} \)
11 \( 1 + (-3.46 - 1.12i)T + (8.89 + 6.46i)T^{2} \)
13 \( 1 + (2.63 - 5.16i)T + (-7.64 - 10.5i)T^{2} \)
17 \( 1 + (0.902 + 0.142i)T + (16.1 + 5.25i)T^{2} \)
19 \( 1 + (-4.29 - 5.91i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + (-3.05 - 6.00i)T + (-13.5 + 18.6i)T^{2} \)
29 \( 1 + (4.30 + 3.12i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.78 - 1.29i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-2.74 - 1.39i)T + (21.7 + 29.9i)T^{2} \)
41 \( 1 + (-2.66 + 0.866i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 + (3.01 + 3.01i)T + 43iT^{2} \)
47 \( 1 + (0.998 + 6.30i)T + (-44.6 + 14.5i)T^{2} \)
53 \( 1 + (8.04 - 1.27i)T + (50.4 - 16.3i)T^{2} \)
59 \( 1 + (-1.98 - 6.10i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (4.21 - 12.9i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-0.976 + 6.16i)T + (-63.7 - 20.7i)T^{2} \)
71 \( 1 + (-5.88 + 8.09i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (8.26 - 4.21i)T + (42.9 - 59.0i)T^{2} \)
79 \( 1 + (-4.29 + 5.90i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (0.190 - 1.20i)T + (-78.9 - 25.6i)T^{2} \)
89 \( 1 + (-3.41 + 10.5i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-3.84 + 0.609i)T + (92.2 - 29.9i)T^{2} \)
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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

−11.63903065234850920046979477506, −11.44581241760197592083336229349, −10.17860771559951253339305485285, −9.182216570701454358330361880349, −7.57123083560437030844662797392, −7.22001800244153980465917768856, −6.12051207697305246887222402969, −4.73158937565011179222709837254, −3.78352098559587992586089968469, −1.61227307357206929733941507787, 0.69755398729664463115506830766, 3.28143688865912131567043954467, 4.62617835156751335588236982302, 5.31722375934568851230747700010, 6.64795099052132928003210021735, 7.67165995095191590299128011410, 8.849274717043607195890977956432, 9.682531317146189565322680357759, 11.07048279823676847699474451253, 11.35447576377901600747831970515

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