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Properties

Label 2-300-75.17-c1-0-2
Degree $2$
Conductor $300$
Sign $0.933 - 0.359i$
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  + (−0.719 − 1.57i)3-s + (−1.13 + 1.92i)5-s + (3.00 + 3.00i)7-s + (−1.96 + 2.26i)9-s + (2.76 − 0.897i)11-s + (0.404 + 0.794i)13-s + (3.85 + 0.405i)15-s + (3.19 − 0.505i)17-s + (0.694 − 0.955i)19-s + (2.57 − 6.90i)21-s + (−2.51 + 4.94i)23-s + (−2.41 − 4.37i)25-s + (4.98 + 1.46i)27-s + (6.77 − 4.92i)29-s + (6.19 + 4.49i)31-s + ⋯
L(s)  = 1  + (−0.415 − 0.909i)3-s + (−0.508 + 0.861i)5-s + (1.13 + 1.13i)7-s + (−0.655 + 0.755i)9-s + (0.832 − 0.270i)11-s + (0.112 + 0.220i)13-s + (0.994 + 0.104i)15-s + (0.774 − 0.122i)17-s + (0.159 − 0.219i)19-s + (0.562 − 1.50i)21-s + (−0.525 + 1.03i)23-s + (−0.483 − 0.875i)25-s + (0.959 + 0.282i)27-s + (1.25 − 0.914i)29-s + (1.11 + 0.808i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 - 0.359i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.933 - 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.933 - 0.359i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.933 - 0.359i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15366 + 0.214444i\)
\(L(\frac12)\) \(\approx\) \(1.15366 + 0.214444i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.719 + 1.57i)T \)
5 \( 1 + (1.13 - 1.92i)T \)
good7 \( 1 + (-3.00 - 3.00i)T + 7iT^{2} \)
11 \( 1 + (-2.76 + 0.897i)T + (8.89 - 6.46i)T^{2} \)
13 \( 1 + (-0.404 - 0.794i)T + (-7.64 + 10.5i)T^{2} \)
17 \( 1 + (-3.19 + 0.505i)T + (16.1 - 5.25i)T^{2} \)
19 \( 1 + (-0.694 + 0.955i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + (2.51 - 4.94i)T + (-13.5 - 18.6i)T^{2} \)
29 \( 1 + (-6.77 + 4.92i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-6.19 - 4.49i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (8.57 - 4.36i)T + (21.7 - 29.9i)T^{2} \)
41 \( 1 + (8.73 + 2.83i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + (1.45 - 1.45i)T - 43iT^{2} \)
47 \( 1 + (1.10 - 6.99i)T + (-44.6 - 14.5i)T^{2} \)
53 \( 1 + (-2.56 - 0.405i)T + (50.4 + 16.3i)T^{2} \)
59 \( 1 + (-3.44 + 10.6i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (2.69 + 8.29i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-0.616 - 3.89i)T + (-63.7 + 20.7i)T^{2} \)
71 \( 1 + (0.265 + 0.365i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (2.68 + 1.36i)T + (42.9 + 59.0i)T^{2} \)
79 \( 1 + (-2.07 - 2.85i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (2.20 + 13.9i)T + (-78.9 + 25.6i)T^{2} \)
89 \( 1 + (2.41 + 7.44i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (1.51 + 0.239i)T + (92.2 + 29.9i)T^{2} \)
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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

−11.78939848479075655400888254187, −11.32497438444401169901322409205, −10.09295377966699842130989844114, −8.570350632371737292849896625177, −7.999455594307933193571654108239, −6.88043464469845829679186146899, −6.00616805220610155391819237012, −4.88312988191563015658151284928, −3.10869892121185513035243550389, −1.68059057208545048693530964371, 1.07943895790893840122405363564, 3.74469438471514152720611146663, 4.47981797749558789700141786503, 5.31915449562598290979112262735, 6.81105277753217714342077500833, 8.086199864650104698863683057913, 8.752398662306601796618450712806, 10.06447285475834144684490072633, 10.62927838268529690656471067944, 11.84087794053973491541048906190

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