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Properties

Label 2-300-25.9-c1-0-3
Degree $2$
Conductor $300$
Sign $0.230 + 0.973i$
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 + (0.971 − 2.01i)5-s − 1.04i·7-s + (0.809 − 0.587i)9-s + (−5.08 − 3.69i)11-s + (0.591 + 0.814i)13-s + (−0.301 + 2.21i)15-s + (4.46 + 1.44i)17-s + (1.84 − 5.68i)19-s + (0.323 + 0.995i)21-s + (4.73 − 6.51i)23-s + (−3.11 − 3.91i)25-s + (−0.587 + 0.809i)27-s + (−2.13 − 6.57i)29-s + (−2.94 + 9.05i)31-s + ⋯
L(s)  = 1  + (−0.549 + 0.178i)3-s + (0.434 − 0.900i)5-s − 0.395i·7-s + (0.269 − 0.195i)9-s + (−1.53 − 1.11i)11-s + (0.164 + 0.225i)13-s + (−0.0778 + 0.572i)15-s + (1.08 + 0.351i)17-s + (0.423 − 1.30i)19-s + (0.0705 + 0.217i)21-s + (0.986 − 1.35i)23-s + (−0.622 − 0.782i)25-s + (−0.113 + 0.155i)27-s + (−0.396 − 1.22i)29-s + (−0.528 + 1.62i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.230 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.230 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.230 + 0.973i$
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: \((2,\ 300,\ (\ :1/2),\ 0.230 + 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.807511 - 0.638874i\)
\(L(\frac12)\) \(\approx\) \(0.807511 - 0.638874i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.971 + 2.01i)T \)
good7 \( 1 + 1.04iT - 7T^{2} \)
11 \( 1 + (5.08 + 3.69i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-0.591 - 0.814i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-4.46 - 1.44i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (-1.84 + 5.68i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-4.73 + 6.51i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (2.13 + 6.57i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (2.94 - 9.05i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-4.52 - 6.22i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (1.26 - 0.922i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 9.94iT - 43T^{2} \)
47 \( 1 + (4.60 - 1.49i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (2.68 - 0.873i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-3.30 + 2.39i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (2.55 + 1.85i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (3.01 + 0.979i)T + (54.2 + 39.3i)T^{2} \)
71 \( 1 + (-2.01 - 6.19i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (0.216 - 0.297i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-1.02 - 3.15i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-13.1 - 4.28i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (2.02 + 1.47i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-0.173 + 0.0564i)T + (78.4 - 57.0i)T^{2} \)
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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

−11.40341101517310621142929311247, −10.65377872542798751236663001988, −9.782621596305223652539620468770, −8.681660865765834241146617282564, −7.82703875118944236005896149431, −6.42778782533791233579084041377, −5.38503488153973882702341538001, −4.67335557460019039697227991938, −2.96949756452317807109542847723, −0.838334159465762442114527160411, 2.03386651222599097434594655349, 3.40531453123452639540475990626, 5.31101373450276301291765092628, 5.74596388470284519171124581174, 7.32452078166703316008280122898, 7.64800672551464850615554623949, 9.427855390396523401791233338870, 10.18124031228737519872947249779, 10.90332078525663943192090440272, 11.94159473173590118725529728612

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