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Properties

Label 2-300-100.27-c1-0-20
Degree $2$
Conductor $300$
Sign $0.838 + 0.544i$
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  + (−1.36 + 0.378i)2-s + (0.891 − 0.453i)3-s + (1.71 − 1.03i)4-s + (2.17 − 0.535i)5-s + (−1.04 + 0.956i)6-s + (2.21 − 2.21i)7-s + (−1.94 + 2.05i)8-s + (0.587 − 0.809i)9-s + (−2.75 + 1.55i)10-s + (−1.46 − 2.01i)11-s + (1.05 − 1.69i)12-s + (−1.47 + 0.233i)13-s + (−2.17 + 3.85i)14-s + (1.69 − 1.46i)15-s + (1.86 − 3.53i)16-s + (−2.58 + 5.08i)17-s + ⋯
L(s)  = 1  + (−0.963 + 0.267i)2-s + (0.514 − 0.262i)3-s + (0.856 − 0.516i)4-s + (0.970 − 0.239i)5-s + (−0.425 + 0.390i)6-s + (0.835 − 0.835i)7-s + (−0.686 + 0.726i)8-s + (0.195 − 0.269i)9-s + (−0.871 + 0.490i)10-s + (−0.441 − 0.607i)11-s + (0.305 − 0.490i)12-s + (−0.409 + 0.0647i)13-s + (−0.581 + 1.02i)14-s + (0.436 − 0.377i)15-s + (0.467 − 0.884i)16-s + (−0.627 + 1.23i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.838 + 0.544i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.838 + 0.544i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.838 + 0.544i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.838 + 0.544i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16077 - 0.343907i\)
\(L(\frac12)\) \(\approx\) \(1.16077 - 0.343907i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 - 0.378i)T \)
3 \( 1 + (-0.891 + 0.453i)T \)
5 \( 1 + (-2.17 + 0.535i)T \)
good7 \( 1 + (-2.21 + 2.21i)T - 7iT^{2} \)
11 \( 1 + (1.46 + 2.01i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (1.47 - 0.233i)T + (12.3 - 4.01i)T^{2} \)
17 \( 1 + (2.58 - 5.08i)T + (-9.99 - 13.7i)T^{2} \)
19 \( 1 + (0.655 + 2.01i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (5.51 + 0.872i)T + (21.8 + 7.10i)T^{2} \)
29 \( 1 + (-7.97 - 2.59i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-4.64 + 1.51i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (-1.30 - 8.25i)T + (-35.1 + 11.4i)T^{2} \)
41 \( 1 + (-1.69 - 1.22i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (2.23 + 2.23i)T + 43iT^{2} \)
47 \( 1 + (-3.04 - 5.98i)T + (-27.6 + 38.0i)T^{2} \)
53 \( 1 + (5.83 + 11.4i)T + (-31.1 + 42.8i)T^{2} \)
59 \( 1 + (-4.78 - 3.47i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (4.94 - 3.58i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-3.66 - 1.86i)T + (39.3 + 54.2i)T^{2} \)
71 \( 1 + (6.22 + 2.02i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (1.32 - 8.37i)T + (-69.4 - 22.5i)T^{2} \)
79 \( 1 + (2.63 - 8.10i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (6.35 - 12.4i)T + (-48.7 - 67.1i)T^{2} \)
89 \( 1 + (5.93 + 8.17i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (-3.50 + 1.78i)T + (57.0 - 78.4i)T^{2} \)
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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

−11.34344272449996681090091925866, −10.41146514181522638044055731821, −9.848679805554161063427414592654, −8.478329532733494672598612561316, −8.218211419205356656615792212193, −6.90090378992204676127592417980, −6.02794555486793184200537563381, −4.62084034535783296753717610088, −2.57105943533000184058584701135, −1.32232667105926298827428381091, 2.00477070970936052317275623328, 2.71206241776597526632087085162, 4.71720234401629671146507539811, 6.01578349510984735065113580371, 7.27818701647045707616621003142, 8.229975755702395871331775009870, 9.114530592728427959587932683151, 9.870164604817191013601435290458, 10.57818989550236393269037987592, 11.71533715535700725739623011745

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