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Properties

Label 2-300-4.3-c2-0-15
Degree $2$
Conductor $300$
Sign $0.514 + 0.857i$
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  + (−0.534 − 1.92i)2-s − 1.73i·3-s + (−3.42 + 2.05i)4-s + (−3.33 + 0.925i)6-s + 11.9i·7-s + (5.79 + 5.51i)8-s − 2.99·9-s − 14.5i·11-s + (3.56 + 5.94i)12-s + 22.4·13-s + (23.0 − 6.39i)14-s + (7.52 − 14.1i)16-s + 12.6·17-s + (1.60 + 5.78i)18-s + 8.76i·19-s + ⋯
L(s)  = 1  + (−0.267 − 0.963i)2-s − 0.577i·3-s + (−0.857 + 0.514i)4-s + (−0.556 + 0.154i)6-s + 1.71i·7-s + (0.724 + 0.688i)8-s − 0.333·9-s − 1.32i·11-s + (0.297 + 0.495i)12-s + 1.72·13-s + (1.64 − 0.456i)14-s + (0.470 − 0.882i)16-s + 0.746·17-s + (0.0890 + 0.321i)18-s + 0.461i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.514 + 0.857i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.514 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.514 + 0.857i$
Analytic conductor: \(8.17440\)
Root analytic conductor: \(2.85909\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1),\ 0.514 + 0.857i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.19626 - 0.677142i\)
\(L(\frac12)\) \(\approx\) \(1.19626 - 0.677142i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.534 + 1.92i)T \)
3 \( 1 + 1.73iT \)
5 \( 1 \)
good7 \( 1 - 11.9iT - 49T^{2} \)
11 \( 1 + 14.5iT - 121T^{2} \)
13 \( 1 - 22.4T + 169T^{2} \)
17 \( 1 - 12.6T + 289T^{2} \)
19 \( 1 - 8.76iT - 361T^{2} \)
23 \( 1 + 4.99iT - 529T^{2} \)
29 \( 1 - 2.74T + 841T^{2} \)
31 \( 1 - 16.3iT - 961T^{2} \)
37 \( 1 - 32.4T + 1.36e3T^{2} \)
41 \( 1 - 42.7T + 1.68e3T^{2} \)
43 \( 1 + 16.5iT - 1.84e3T^{2} \)
47 \( 1 - 48.5iT - 2.20e3T^{2} \)
53 \( 1 - 94.1T + 2.80e3T^{2} \)
59 \( 1 + 43.2iT - 3.48e3T^{2} \)
61 \( 1 - 56.7T + 3.72e3T^{2} \)
67 \( 1 + 61.1iT - 4.48e3T^{2} \)
71 \( 1 + 39.6iT - 5.04e3T^{2} \)
73 \( 1 + 99.5T + 5.32e3T^{2} \)
79 \( 1 - 10.7iT - 6.24e3T^{2} \)
83 \( 1 - 140. iT - 6.88e3T^{2} \)
89 \( 1 - 54.8T + 7.92e3T^{2} \)
97 \( 1 - 14.1T + 9.40e3T^{2} \)
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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

−11.44459855663201574636547984777, −10.72233279609414034000887144979, −9.309122130735938949319098211786, −8.582170690195476535305569755477, −8.058113165538913876855276918135, −6.14726298091961183278218554708, −5.49683947177391183730078399492, −3.61796906999894973479839425140, −2.60029944476819899145037096747, −1.14132465030501480292308923314, 1.02052608948419132526273470163, 3.83307896495019608215957845310, 4.45353987570142024503241733034, 5.80381633022608374044439941485, 6.96677448602879162102702632442, 7.67189101078367121170007584804, 8.767697113425463068902351931279, 9.886099956260114833027360722458, 10.36952473203679723048397597725, 11.36249908469139262964568163019

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