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Properties

Label 2-300-20.3-c1-0-1
Degree $2$
Conductor $300$
Sign $-0.755 - 0.655i$
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  + (−0.707 + 1.22i)2-s + (−0.707 + 0.707i)3-s + (−0.999 − 1.73i)4-s + (−0.366 − 1.36i)6-s + (3.15 + 3.15i)7-s + 2.82·8-s − 1.00i·9-s + 2i·11-s + (1.93 + 0.517i)12-s + (−1.60 − 1.60i)13-s + (−6.09 + 1.63i)14-s + (−2.00 + 3.46i)16-s + (−4.24 + 4.24i)17-s + (1.22 + 0.707i)18-s + 3.19·19-s + ⋯
L(s)  = 1  + (−0.499 + 0.866i)2-s + (−0.408 + 0.408i)3-s + (−0.499 − 0.866i)4-s + (−0.149 − 0.557i)6-s + (1.19 + 1.19i)7-s + 0.999·8-s − 0.333i·9-s + 0.603i·11-s + (0.557 + 0.149i)12-s + (−0.444 − 0.444i)13-s + (−1.62 + 0.436i)14-s + (−0.500 + 0.866i)16-s + (−1.02 + 1.02i)17-s + (0.288 + 0.166i)18-s + 0.733·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 - 0.655i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.755 - 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.755 - 0.655i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ -0.755 - 0.655i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.292321 + 0.782911i\)
\(L(\frac12)\) \(\approx\) \(0.292321 + 0.782911i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 1.22i)T \)
3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 \)
good7 \( 1 + (-3.15 - 3.15i)T + 7iT^{2} \)
11 \( 1 - 2iT - 11T^{2} \)
13 \( 1 + (1.60 + 1.60i)T + 13iT^{2} \)
17 \( 1 + (4.24 - 4.24i)T - 17iT^{2} \)
19 \( 1 - 3.19T + 19T^{2} \)
23 \( 1 + (5.27 - 5.27i)T - 23iT^{2} \)
29 \( 1 + 0.535iT - 29T^{2} \)
31 \( 1 - 3.73iT - 31T^{2} \)
37 \( 1 + (-7.72 + 7.72i)T - 37iT^{2} \)
41 \( 1 - 1.46T + 41T^{2} \)
43 \( 1 + (3.15 - 3.15i)T - 43iT^{2} \)
47 \( 1 + (0.656 + 0.656i)T + 47iT^{2} \)
53 \( 1 + (-3.86 - 3.86i)T + 53iT^{2} \)
59 \( 1 - 11.4T + 59T^{2} \)
61 \( 1 - 3T + 61T^{2} \)
67 \( 1 + (3.91 + 3.91i)T + 67iT^{2} \)
71 \( 1 + 2.53iT - 71T^{2} \)
73 \( 1 + (-2.82 - 2.82i)T + 73iT^{2} \)
79 \( 1 + 7.46T + 79T^{2} \)
83 \( 1 + (-9.14 + 9.14i)T - 83iT^{2} \)
89 \( 1 + 6.92iT - 89T^{2} \)
97 \( 1 + (-6.88 + 6.88i)T - 97iT^{2} \)
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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

−11.89751070121739482745073009571, −11.08480740080487544586750004581, −10.05489461205211819124051959137, −9.170373605603036857191866796821, −8.283902235055304522476029838913, −7.42174811574205382612199055818, −6.01885798963347383938639929712, −5.31778834999947514289019944709, −4.34424633014078671231037675409, −1.92753299745180275473370532683, 0.803625331949987604251320135723, 2.31723318336624174704414250190, 4.10618377130692372675998514176, 4.96095121256671559131796653808, 6.77830035967490263310000599077, 7.69430605405157103261854784641, 8.460345151013375483154990769837, 9.732161559996905788141181637893, 10.62741591981979422901082805113, 11.48665803916868616207868375247

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