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Properties

Label 2-300-75.53-c1-0-3
Degree $2$
Conductor $300$
Sign $0.867 - 0.498i$
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  + (1.66 − 0.468i)3-s + (−0.354 + 2.20i)5-s + (−1.67 + 1.67i)7-s + (2.56 − 1.56i)9-s + (5.27 + 1.71i)11-s + (−1.00 + 1.96i)13-s + (0.444 + 3.84i)15-s + (4.25 + 0.674i)17-s + (−4.59 − 6.31i)19-s + (−2.00 + 3.56i)21-s + (−1.53 − 3.01i)23-s + (−4.74 − 1.56i)25-s + (3.53 − 3.80i)27-s + (−0.409 − 0.297i)29-s + (2.41 − 1.75i)31-s + ⋯
L(s)  = 1  + (0.962 − 0.270i)3-s + (−0.158 + 0.987i)5-s + (−0.631 + 0.631i)7-s + (0.853 − 0.521i)9-s + (1.59 + 0.517i)11-s + (−0.277 + 0.545i)13-s + (0.114 + 0.993i)15-s + (1.03 + 0.163i)17-s + (−1.05 − 1.44i)19-s + (−0.436 + 0.778i)21-s + (−0.320 − 0.628i)23-s + (−0.949 − 0.312i)25-s + (0.680 − 0.732i)27-s + (−0.0759 − 0.0552i)29-s + (0.434 − 0.315i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.867 - 0.498i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.867 - 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.867 - 0.498i$
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: \((2,\ 300,\ (\ :1/2),\ 0.867 - 0.498i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.64633 + 0.439154i\)
\(L(\frac12)\) \(\approx\) \(1.64633 + 0.439154i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.66 + 0.468i)T \)
5 \( 1 + (0.354 - 2.20i)T \)
good7 \( 1 + (1.67 - 1.67i)T - 7iT^{2} \)
11 \( 1 + (-5.27 - 1.71i)T + (8.89 + 6.46i)T^{2} \)
13 \( 1 + (1.00 - 1.96i)T + (-7.64 - 10.5i)T^{2} \)
17 \( 1 + (-4.25 - 0.674i)T + (16.1 + 5.25i)T^{2} \)
19 \( 1 + (4.59 + 6.31i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + (1.53 + 3.01i)T + (-13.5 + 18.6i)T^{2} \)
29 \( 1 + (0.409 + 0.297i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-2.41 + 1.75i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (7.68 + 3.91i)T + (21.7 + 29.9i)T^{2} \)
41 \( 1 + (4.48 - 1.45i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 + (-5.88 - 5.88i)T + 43iT^{2} \)
47 \( 1 + (-0.527 - 3.33i)T + (-44.6 + 14.5i)T^{2} \)
53 \( 1 + (5.21 - 0.825i)T + (50.4 - 16.3i)T^{2} \)
59 \( 1 + (3.17 + 9.76i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-1.65 + 5.10i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-0.707 + 4.46i)T + (-63.7 - 20.7i)T^{2} \)
71 \( 1 + (-6.04 + 8.32i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (1.93 - 0.983i)T + (42.9 - 59.0i)T^{2} \)
79 \( 1 + (-10.1 + 13.9i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (1.93 - 12.2i)T + (-78.9 - 25.6i)T^{2} \)
89 \( 1 + (1.43 - 4.41i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (15.9 - 2.53i)T + (92.2 - 29.9i)T^{2} \)
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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

−12.04825634148643621098430753066, −10.85628703970737311198696478030, −9.609494408232804700088077523917, −9.188963679822591162804040228562, −7.981767857543450317367558419985, −6.78076912810963939489608364897, −6.42752946825642401920335859356, −4.32275438908824218553397376854, −3.24053343551190906261745575296, −2.09977753885511064238464645357, 1.41866961160740141956319089058, 3.50321370564761104974291763055, 4.08296687178131494836611298777, 5.59549058064732708509003461586, 6.94822369793789451783937869328, 8.085409336964442419614502435638, 8.776516639188682743942868440548, 9.740969954305813610500682821580, 10.36970888484567493293210038263, 11.95520835422208053217012043033

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