Properties

Label 2-300-25.9-c1-0-2
Degree $2$
Conductor $300$
Sign $0.938 + 0.345i$
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 + (2.23 − 0.0974i)5-s − 1.31i·7-s + (0.809 − 0.587i)9-s + (−1.25 − 0.913i)11-s + (1.42 + 1.96i)13-s + (2.09 − 0.782i)15-s + (−1.25 − 0.406i)17-s + (0.315 − 0.971i)19-s + (−0.407 − 1.25i)21-s + (−2.94 + 4.05i)23-s + (4.98 − 0.435i)25-s + (0.587 − 0.809i)27-s + (1.82 + 5.61i)29-s + (2.73 − 8.41i)31-s + ⋯
L(s)  = 1  + (0.549 − 0.178i)3-s + (0.999 − 0.0435i)5-s − 0.498i·7-s + (0.269 − 0.195i)9-s + (−0.379 − 0.275i)11-s + (0.395 + 0.544i)13-s + (0.540 − 0.202i)15-s + (−0.303 − 0.0985i)17-s + (0.0723 − 0.222i)19-s + (−0.0889 − 0.273i)21-s + (−0.613 + 0.844i)23-s + (0.996 − 0.0870i)25-s + (0.113 − 0.155i)27-s + (0.338 + 1.04i)29-s + (0.491 − 1.51i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 + 0.345i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.938 + 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.938 + 0.345i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.938 + 0.345i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.72145 - 0.306731i\)
\(L(\frac12)\) \(\approx\) \(1.72145 - 0.306731i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.23 + 0.0974i)T \)
good7 \( 1 + 1.31iT - 7T^{2} \)
11 \( 1 + (1.25 + 0.913i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-1.42 - 1.96i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.25 + 0.406i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (-0.315 + 0.971i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (2.94 - 4.05i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (-1.82 - 5.61i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-2.73 + 8.41i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (2.95 + 4.06i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (6.43 - 4.67i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 6.84iT - 43T^{2} \)
47 \( 1 + (7.37 - 2.39i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (3.75 - 1.22i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (6.35 - 4.61i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-2.83 - 2.05i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (7.92 + 2.57i)T + (54.2 + 39.3i)T^{2} \)
71 \( 1 + (4.00 + 12.3i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (7.47 - 10.2i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-0.386 - 1.18i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (7.80 + 2.53i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (-7.74 - 5.62i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (15.4 - 5.02i)T + (78.4 - 57.0i)T^{2} \)
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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

−11.64553176850661424260614333554, −10.62965585574861497340661598837, −9.727919335804390055058005735540, −8.954555105505838521566129004680, −7.891197533321847021772082519756, −6.79900087482997058150649406434, −5.82342423897237659863498796213, −4.48382865385981964113808468161, −3.04074976899952369143935965326, −1.62971731030974435480704259421, 1.95999729966454278979058544130, 3.12998570566846868476128448405, 4.74577587145520240758053276057, 5.81375012472657840532456203528, 6.84441482001603456720328655706, 8.255477612973095427462171235377, 8.876522770944674427288517424722, 10.08613719036699661613602992733, 10.44709684572760166209795070710, 11.91779991795138463487640252486

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