Properties

Label 2-300-300.131-c1-0-38
Degree $2$
Conductor $300$
Sign $0.205 + 0.978i$
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  + (−0.322 − 1.37i)2-s + (1.63 + 0.576i)3-s + (−1.79 + 0.889i)4-s + (1.79 − 1.33i)5-s + (0.266 − 2.43i)6-s − 3.04i·7-s + (1.80 + 2.17i)8-s + (2.33 + 1.88i)9-s + (−2.41 − 2.03i)10-s + (−0.519 − 1.59i)11-s + (−3.43 + 0.419i)12-s + (−1.88 + 5.80i)13-s + (−4.18 + 0.982i)14-s + (3.69 − 1.14i)15-s + (2.41 − 3.18i)16-s + (0.841 − 1.15i)17-s + ⋯
L(s)  = 1  + (−0.228 − 0.973i)2-s + (0.942 + 0.332i)3-s + (−0.895 + 0.444i)4-s + (0.802 − 0.596i)5-s + (0.108 − 0.994i)6-s − 1.14i·7-s + (0.637 + 0.770i)8-s + (0.778 + 0.627i)9-s + (−0.764 − 0.644i)10-s + (−0.156 − 0.482i)11-s + (−0.992 + 0.121i)12-s + (−0.523 + 1.61i)13-s + (−1.11 + 0.262i)14-s + (0.955 − 0.295i)15-s + (0.604 − 0.796i)16-s + (0.204 − 0.280i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.205 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.205 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.205 + 0.978i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.205 + 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24291 - 1.00890i\)
\(L(\frac12)\) \(\approx\) \(1.24291 - 1.00890i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.322 + 1.37i)T \)
3 \( 1 + (-1.63 - 0.576i)T \)
5 \( 1 + (-1.79 + 1.33i)T \)
good7 \( 1 + 3.04iT - 7T^{2} \)
11 \( 1 + (0.519 + 1.59i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (1.88 - 5.80i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.841 + 1.15i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (-2.14 + 2.95i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + (1.33 + 4.09i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (1.32 + 1.82i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (3.80 - 5.23i)T + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (3.25 - 10.0i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-8.40 - 2.73i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 - 2.84iT - 43T^{2} \)
47 \( 1 + (5.06 - 3.68i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (5.69 + 7.84i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (3.64 - 11.2i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-0.190 - 0.587i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (1.22 - 1.68i)T + (-20.7 - 63.7i)T^{2} \)
71 \( 1 + (0.331 - 0.240i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.657 - 2.02i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-8.14 - 11.2i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (5.79 + 4.21i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-0.333 + 0.108i)T + (72.0 - 52.3i)T^{2} \)
97 \( 1 + (0.649 - 0.471i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.37921158191867813158671480609, −10.39280040061356602226959121451, −9.639939876031965401090414781202, −9.046451312417437886922538991050, −8.066358098948872098897674116220, −6.89805886411007069481080595382, −4.90085596210714006349099129046, −4.18753100029576222861962116952, −2.80369531752899381938096467290, −1.45482000153911723310418193684, 2.05078779193639759307941200854, 3.43505728389450286991955260590, 5.37531375913645572999408410031, 5.99054871158270037691823085768, 7.37863836393864953657807580428, 7.898045516928728563770568026508, 9.136412349923904493963837158527, 9.662724529742743067292780708774, 10.58230975796756051216145940204, 12.44553994053021612911891771570

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