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Properties

Label 2-300-300.179-c1-0-36
Degree $2$
Conductor $300$
Sign $0.621 - 0.783i$
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  + (0.665 + 1.24i)2-s + (1.31 − 1.12i)3-s + (−1.11 + 1.66i)4-s + (2.18 + 0.455i)5-s + (2.28 + 0.892i)6-s − 0.246·7-s + (−2.81 − 0.285i)8-s + (0.463 − 2.96i)9-s + (0.888 + 3.03i)10-s + (−1.26 + 3.90i)11-s + (0.403 + 3.44i)12-s + (3.99 − 1.29i)13-s + (−0.164 − 0.307i)14-s + (3.39 − 1.86i)15-s + (−1.51 − 3.70i)16-s + (−1.31 + 0.954i)17-s + ⋯
L(s)  = 1  + (0.470 + 0.882i)2-s + (0.759 − 0.650i)3-s + (−0.557 + 0.830i)4-s + (0.979 + 0.203i)5-s + (0.931 + 0.364i)6-s − 0.0931·7-s + (−0.994 − 0.100i)8-s + (0.154 − 0.987i)9-s + (0.280 + 0.959i)10-s + (−0.382 + 1.17i)11-s + (0.116 + 0.993i)12-s + (1.10 − 0.359i)13-s + (−0.0438 − 0.0822i)14-s + (0.876 − 0.481i)15-s + (−0.379 − 0.925i)16-s + (−0.318 + 0.231i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.621 - 0.783i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.621 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.621 - 0.783i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.621 - 0.783i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.92180 + 0.928907i\)
\(L(\frac12)\) \(\approx\) \(1.92180 + 0.928907i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.665 - 1.24i)T \)
3 \( 1 + (-1.31 + 1.12i)T \)
5 \( 1 + (-2.18 - 0.455i)T \)
good7 \( 1 + 0.246T + 7T^{2} \)
11 \( 1 + (1.26 - 3.90i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-3.99 + 1.29i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.31 - 0.954i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (3.07 + 4.23i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + (0.275 + 0.0896i)T + (18.6 + 13.5i)T^{2} \)
29 \( 1 + (3.35 - 4.61i)T + (-8.96 - 27.5i)T^{2} \)
31 \( 1 + (2.15 + 2.96i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (3.54 - 1.15i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-6.74 + 2.19i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 + 5.82T + 43T^{2} \)
47 \( 1 + (1.31 - 1.80i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (6.78 + 4.93i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-1.91 - 5.87i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-3.54 + 10.9i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-5.77 + 4.19i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (12.8 + 9.35i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-5.72 - 1.85i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-5.94 + 8.17i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (-10.0 - 13.8i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + (-6.78 - 2.20i)T + (72.0 + 52.3i)T^{2} \)
97 \( 1 + (10.0 - 13.7i)T + (-29.9 - 92.2i)T^{2} \)
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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

−12.47814541437040360569703120935, −10.93813115422259123564408426162, −9.601971950566241260013289811850, −8.896790155017873876531782473828, −7.898402988419719477244253624657, −6.86436817423233933995073685704, −6.23716083656895005783617201183, −4.94664592933379636181178739478, −3.47999700913433487780067778886, −2.12645392870468343522976983454, 1.79765721125305987378575355770, 3.09175847773372497511225467934, 4.13109731364490015306772252839, 5.41696083978087891338383709733, 6.24599087892954577693192494794, 8.321267977351043932653470849178, 8.961074196566697608277253748719, 9.858269729528746308948225228539, 10.66028441294112253165680828376, 11.34491670486414012257778798565

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