Properties

Label 2-300-20.19-c2-0-11
Degree $2$
Conductor $300$
Sign $-0.544 - 0.838i$
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  + (1.49 + 1.33i)2-s − 1.73·3-s + (0.446 + 3.97i)4-s + (−2.58 − 2.30i)6-s + 6.56·7-s + (−4.63 + 6.52i)8-s + 2.99·9-s + 2.26i·11-s + (−0.773 − 6.88i)12-s + 14.8i·13-s + (9.79 + 8.75i)14-s + (−15.6 + 3.55i)16-s + 26.8i·17-s + (4.47 + 3.99i)18-s − 10.8i·19-s + ⋯
L(s)  = 1  + (0.745 + 0.666i)2-s − 0.577·3-s + (0.111 + 0.993i)4-s + (−0.430 − 0.384i)6-s + 0.938·7-s + (−0.579 + 0.815i)8-s + 0.333·9-s + 0.206i·11-s + (−0.0644 − 0.573i)12-s + 1.14i·13-s + (0.699 + 0.625i)14-s + (−0.975 + 0.221i)16-s + 1.57i·17-s + (0.248 + 0.222i)18-s − 0.572i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.544 - 0.838i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.544 - 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.544 - 0.838i$
Analytic conductor: \(8.17440\)
Root analytic conductor: \(2.85909\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1),\ -0.544 - 0.838i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.945064 + 1.73978i\)
\(L(\frac12)\) \(\approx\) \(0.945064 + 1.73978i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.49 - 1.33i)T \)
3 \( 1 + 1.73T \)
5 \( 1 \)
good7 \( 1 - 6.56T + 49T^{2} \)
11 \( 1 - 2.26iT - 121T^{2} \)
13 \( 1 - 14.8iT - 169T^{2} \)
17 \( 1 - 26.8iT - 289T^{2} \)
19 \( 1 + 10.8iT - 361T^{2} \)
23 \( 1 + 36.4T + 529T^{2} \)
29 \( 1 - 35.2T + 841T^{2} \)
31 \( 1 + 23.8iT - 961T^{2} \)
37 \( 1 - 54.7iT - 1.36e3T^{2} \)
41 \( 1 + 23.8T + 1.68e3T^{2} \)
43 \( 1 - 56.2T + 1.84e3T^{2} \)
47 \( 1 - 51.4T + 2.20e3T^{2} \)
53 \( 1 + 30.6iT - 2.80e3T^{2} \)
59 \( 1 - 6.92iT - 3.48e3T^{2} \)
61 \( 1 - 107.T + 3.72e3T^{2} \)
67 \( 1 - 111.T + 4.48e3T^{2} \)
71 \( 1 + 31.3iT - 5.04e3T^{2} \)
73 \( 1 + 110. iT - 5.32e3T^{2} \)
79 \( 1 + 59.0iT - 6.24e3T^{2} \)
83 \( 1 + 142.T + 6.88e3T^{2} \)
89 \( 1 + 7.14T + 7.92e3T^{2} \)
97 \( 1 + 126. iT - 9.40e3T^{2} \)
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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

−11.87216470455480156655291330363, −11.28060031443392537696017347764, −10.10032198930686401734113657013, −8.649654392765441313456845666777, −7.88785372747091348107671077057, −6.71674803542116237045005790065, −5.93618308347575865481875046039, −4.72709123963314347056759610757, −4.01488241433500516672555444102, −2.00743250199212259775634919645, 0.844756334688072721468892423462, 2.48382794445740902920922119458, 3.99526079792327188882269347064, 5.13524152318509295184014037857, 5.78322361393595148191829535919, 7.13626931753170126415105113903, 8.324507887811332051670493734970, 9.739232905724475572204381895615, 10.52913472388915936852309364189, 11.32929216202213817631590467755

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