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Properties

Label 2-300-25.14-c1-0-0
Degree $2$
Conductor $300$
Sign $-0.969 - 0.243i$
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  + (−0.951 − 0.309i)3-s + (−2.10 − 0.751i)5-s − 0.595i·7-s + (0.809 + 0.587i)9-s + (−2.71 + 1.97i)11-s + (−2.80 + 3.85i)13-s + (1.77 + 1.36i)15-s + (−7.11 + 2.31i)17-s + (−1.91 − 5.88i)19-s + (−0.184 + 0.566i)21-s + (−2.59 − 3.56i)23-s + (3.86 + 3.16i)25-s + (−0.587 − 0.809i)27-s + (0.853 − 2.62i)29-s + (1.38 + 4.26i)31-s + ⋯
L(s)  = 1  + (−0.549 − 0.178i)3-s + (−0.941 − 0.336i)5-s − 0.225i·7-s + (0.269 + 0.195i)9-s + (−0.818 + 0.594i)11-s + (−0.777 + 1.07i)13-s + (0.457 + 0.352i)15-s + (−1.72 + 0.560i)17-s + (−0.438 − 1.35i)19-s + (−0.0401 + 0.123i)21-s + (−0.540 − 0.743i)23-s + (0.773 + 0.633i)25-s + (−0.113 − 0.155i)27-s + (0.158 − 0.487i)29-s + (0.249 + 0.766i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 - 0.243i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.969 - 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.969 - 0.243i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ -0.969 - 0.243i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00164505 + 0.0133127i\)
\(L(\frac12)\) \(\approx\) \(0.00164505 + 0.0133127i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.951 + 0.309i)T \)
5 \( 1 + (2.10 + 0.751i)T \)
good7 \( 1 + 0.595iT - 7T^{2} \)
11 \( 1 + (2.71 - 1.97i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (2.80 - 3.85i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (7.11 - 2.31i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (1.91 + 5.88i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (2.59 + 3.56i)T + (-7.10 + 21.8i)T^{2} \)
29 \( 1 + (-0.853 + 2.62i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-1.38 - 4.26i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (0.764 - 1.05i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (-7.61 - 5.53i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 7.59iT - 43T^{2} \)
47 \( 1 + (4.18 + 1.36i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (7.80 + 2.53i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (1.80 + 1.31i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (10.2 - 7.41i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-7.94 + 2.58i)T + (54.2 - 39.3i)T^{2} \)
71 \( 1 + (-2.09 + 6.46i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-4.29 - 5.91i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-3.26 + 10.0i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (3.97 - 1.29i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + (-3.43 + 2.49i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (11.6 + 3.79i)T + (78.4 + 57.0i)T^{2} \)
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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

−12.17643265162473178599077171507, −11.24301487217401408527265410154, −10.58445346110039926136323507430, −9.289079661943230968285871806399, −8.317599267403757260272698629575, −7.21275936537405476358931838603, −6.53398453338757781374395099193, −4.77639909851334346754098370261, −4.36057784882285394181282142041, −2.31349371031181170983552900028, 0.009852337676947862238916812008, 2.71374828515130514654192387662, 4.07540832510679570218000839675, 5.23583605728403848957817122421, 6.30436671099497448903164342358, 7.55876438058656540169843363874, 8.225169222738149293378249291831, 9.557563791540171472802052355489, 10.68394604540929365044080530167, 11.14683446180788361357776429685

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