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Properties

Label 2-300-25.4-c1-0-2
Degree $2$
Conductor $300$
Sign $0.647 - 0.761i$
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  + (−0.587 + 0.809i)3-s + (1.99 − 1.00i)5-s + 3.80i·7-s + (−0.309 − 0.951i)9-s + (0.0589 − 0.181i)11-s + (1.59 − 0.518i)13-s + (−0.364 + 2.20i)15-s + (2.70 + 3.72i)17-s + (−2.13 + 1.55i)19-s + (−3.08 − 2.23i)21-s + (6.04 + 1.96i)23-s + (2.99 − 4.00i)25-s + (0.951 + 0.309i)27-s + (2.03 + 1.48i)29-s + (−3.03 + 2.20i)31-s + ⋯
L(s)  = 1  + (−0.339 + 0.467i)3-s + (0.894 − 0.447i)5-s + 1.44i·7-s + (−0.103 − 0.317i)9-s + (0.0177 − 0.0546i)11-s + (0.442 − 0.143i)13-s + (−0.0941 + 0.569i)15-s + (0.656 + 0.903i)17-s + (−0.490 + 0.356i)19-s + (−0.672 − 0.488i)21-s + (1.26 + 0.409i)23-s + (0.598 − 0.800i)25-s + (0.183 + 0.0594i)27-s + (0.378 + 0.275i)29-s + (−0.544 + 0.395i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.647 - 0.761i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.647 - 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.647 - 0.761i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.647 - 0.761i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21950 + 0.563754i\)
\(L(\frac12)\) \(\approx\) \(1.21950 + 0.563754i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.587 - 0.809i)T \)
5 \( 1 + (-1.99 + 1.00i)T \)
good7 \( 1 - 3.80iT - 7T^{2} \)
11 \( 1 + (-0.0589 + 0.181i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-1.59 + 0.518i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-2.70 - 3.72i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (2.13 - 1.55i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-6.04 - 1.96i)T + (18.6 + 13.5i)T^{2} \)
29 \( 1 + (-2.03 - 1.48i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (3.03 - 2.20i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (11.2 - 3.66i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.22 + 6.83i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 9.22iT - 43T^{2} \)
47 \( 1 + (-2.67 + 3.67i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-5.54 + 7.62i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.20 + 6.79i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-2.94 + 9.06i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-3.55 - 4.89i)T + (-20.7 + 63.7i)T^{2} \)
71 \( 1 + (10.7 + 7.81i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (4.95 + 1.61i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (2.51 + 1.82i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (2.74 + 3.78i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + (4.30 - 13.2i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (3.93 - 5.41i)T + (-29.9 - 92.2i)T^{2} \)
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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

−12.06413159765460737885680117932, −10.78707610423099111244846010451, −10.04316278779687449536159792639, −8.885087945851864322226036971176, −8.572157434049827648703615465956, −6.70852661012102210969502482963, −5.61593500390452703133920376196, −5.19822675763699456639745800361, −3.44619116788592717879708690091, −1.85452482320152759678741850525, 1.20863919565543134581840099675, 2.95049352793426326532030889737, 4.51002190623006212863555960899, 5.76599377377013568076463332254, 6.88487357052786078765613958386, 7.36269739611383015699529486001, 8.828099523276115972282501084066, 9.965668564531742598105807966870, 10.68820869121903971202526825332, 11.41087129268668687496921430041

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