Properties

Label 2-300-3.2-c2-0-9
Degree $2$
Conductor $300$
Sign $-0.666 + 0.745i$
Analytic cond. $8.17440$
Root an. cond. $2.85909$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2 + 2.23i)3-s − 2·7-s + (−1.00 − 8.94i)9-s + 13.4i·11-s − 8·13-s − 13.4i·17-s − 34·19-s + (4 − 4.47i)21-s − 40.2i·23-s + (22.0 + 15.6i)27-s − 40.2i·29-s + 14·31-s + (−30.0 − 26.8i)33-s − 56·37-s + (16 − 17.8i)39-s + ⋯
L(s)  = 1  + (−0.666 + 0.745i)3-s − 0.285·7-s + (−0.111 − 0.993i)9-s + 1.21i·11-s − 0.615·13-s − 0.789i·17-s − 1.78·19-s + (0.190 − 0.212i)21-s − 1.74i·23-s + (0.814 + 0.579i)27-s − 1.38i·29-s + 0.451·31-s + (−0.909 − 0.813i)33-s − 1.51·37-s + (0.410 − 0.458i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.666 + 0.745i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.666 + 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.666 + 0.745i$
Analytic conductor: \(8.17440\)
Root analytic conductor: \(2.85909\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1),\ -0.666 + 0.745i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0495787 - 0.110861i\)
\(L(\frac12)\) \(\approx\) \(0.0495787 - 0.110861i\)
\(L(2)\) 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 \( 1 + (2 - 2.23i)T \)
5 \( 1 \)
good7 \( 1 + 2T + 49T^{2} \)
11 \( 1 - 13.4iT - 121T^{2} \)
13 \( 1 + 8T + 169T^{2} \)
17 \( 1 + 13.4iT - 289T^{2} \)
19 \( 1 + 34T + 361T^{2} \)
23 \( 1 + 40.2iT - 529T^{2} \)
29 \( 1 + 40.2iT - 841T^{2} \)
31 \( 1 - 14T + 961T^{2} \)
37 \( 1 + 56T + 1.36e3T^{2} \)
41 \( 1 + 26.8iT - 1.68e3T^{2} \)
43 \( 1 + 8T + 1.84e3T^{2} \)
47 \( 1 - 40.2iT - 2.20e3T^{2} \)
53 \( 1 - 40.2iT - 2.80e3T^{2} \)
59 \( 1 - 13.4iT - 3.48e3T^{2} \)
61 \( 1 + 46T + 3.72e3T^{2} \)
67 \( 1 + 32T + 4.48e3T^{2} \)
71 \( 1 - 53.6iT - 5.04e3T^{2} \)
73 \( 1 - 106T + 5.32e3T^{2} \)
79 \( 1 + 22T + 6.24e3T^{2} \)
83 \( 1 - 120. iT - 6.88e3T^{2} \)
89 \( 1 - 107. iT - 7.92e3T^{2} \)
97 \( 1 + 122T + 9.40e3T^{2} \)
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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

−11.03348689945468347922973209114, −10.21864449823161517679693539571, −9.565093579568376055727374681660, −8.476213058343883407330887129666, −7.03090524754265786503083261382, −6.24466566045780383254682816019, −4.85754041127780784443162331924, −4.21209483508556133464090483093, −2.46400498127715373100561176813, −0.06113377302895533249386842189, 1.73456102804551241800635838660, 3.40651597295852309092673409727, 5.03066604984611182841641545587, 6.05027237244005334117503226722, 6.83877511425849965420923513757, 7.999983778812660835548324606786, 8.845963533761795134348908928635, 10.27625407967220984240041654128, 10.99188207958482789277875478884, 11.87864734125705491837559771716

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