Properties

Label 2-300-300.131-c1-0-13
Degree $2$
Conductor $300$
Sign $-0.630 - 0.776i$
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  + (−0.446 + 1.34i)2-s + (1.72 + 0.175i)3-s + (−1.60 − 1.19i)4-s + (−2.16 − 0.575i)5-s + (−1.00 + 2.23i)6-s + 4.70i·7-s + (2.32 − 1.61i)8-s + (2.93 + 0.603i)9-s + (1.73 − 2.64i)10-s + (0.0957 + 0.294i)11-s + (−2.54 − 2.34i)12-s + (−1.61 + 4.96i)13-s + (−6.31 − 2.10i)14-s + (−3.62 − 1.37i)15-s + (1.12 + 3.83i)16-s + (−2.69 + 3.70i)17-s + ⋯
L(s)  = 1  + (−0.316 + 0.948i)2-s + (0.994 + 0.101i)3-s + (−0.800 − 0.599i)4-s + (−0.966 − 0.257i)5-s + (−0.410 + 0.911i)6-s + 1.77i·7-s + (0.821 − 0.569i)8-s + (0.979 + 0.201i)9-s + (0.549 − 0.835i)10-s + (0.0288 + 0.0888i)11-s + (−0.735 − 0.677i)12-s + (−0.447 + 1.37i)13-s + (−1.68 − 0.562i)14-s + (−0.935 − 0.353i)15-s + (0.280 + 0.959i)16-s + (−0.652 + 0.898i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.630 - 0.776i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.630 - 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.630 - 0.776i$
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: \((2,\ 300,\ (\ :1/2),\ -0.630 - 0.776i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.491532 + 1.03282i\)
\(L(\frac12)\) \(\approx\) \(0.491532 + 1.03282i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.446 - 1.34i)T \)
3 \( 1 + (-1.72 - 0.175i)T \)
5 \( 1 + (2.16 + 0.575i)T \)
good7 \( 1 - 4.70iT - 7T^{2} \)
11 \( 1 + (-0.0957 - 0.294i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (1.61 - 4.96i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (2.69 - 3.70i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (-2.88 + 3.96i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + (0.222 + 0.685i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (2.34 + 3.22i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (-2.15 + 2.96i)T + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.08 + 3.33i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-6.26 - 2.03i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + 2.91iT - 43T^{2} \)
47 \( 1 + (-3.23 + 2.34i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-1.65 - 2.27i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.546 + 1.68i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (1.48 + 4.58i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-6.49 + 8.93i)T + (-20.7 - 63.7i)T^{2} \)
71 \( 1 + (-0.155 + 0.112i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.448 - 1.37i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-6.71 - 9.24i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (3.47 + 2.52i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (4.46 - 1.45i)T + (72.0 - 52.3i)T^{2} \)
97 \( 1 + (-2.12 + 1.54i)T + (29.9 - 92.2i)T^{2} \)
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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

−12.21754006259271653611400362092, −11.17278073085229349347586085137, −9.523607352325275517944270642914, −9.057561364163566823823605476081, −8.375476588598671460357888342188, −7.46470572834327154998951560488, −6.39727901856042576045944904269, −4.98441808772326980016911670424, −4.02561014541920868170067426631, −2.24462519405955405503425075681, 0.902931513946712535928834889299, 2.95690335595123037733671021663, 3.73076583259114455039316643129, 4.67546633049729044792887075375, 7.25990032753822983629741062133, 7.59593461274967317807641295222, 8.498735746437582539612454276623, 9.782725356678378726281636891992, 10.41023311264644972718360659306, 11.22354777251255499012354657892

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