Properties

Label 2-300-15.14-c2-0-11
Degree $2$
Conductor $300$
Sign $-0.894 + 0.447i$
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  − 3i·3-s − 2i·7-s − 9·9-s − 22i·13-s − 26·19-s − 6·21-s + 27i·27-s − 46·31-s − 26i·37-s − 66·39-s − 22i·43-s + 45·49-s + 78i·57-s + 74·61-s + 18i·63-s + ⋯
L(s)  = 1  i·3-s − 0.285i·7-s − 9-s − 1.69i·13-s − 1.36·19-s − 0.285·21-s + i·27-s − 1.48·31-s − 0.702i·37-s − 1.69·39-s − 0.511i·43-s + 0.918·49-s + 1.36i·57-s + 1.21·61-s + 0.285i·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(8.17440\)
Root analytic conductor: \(2.85909\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1),\ -0.894 + 0.447i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.239920 - 1.01631i\)
\(L(\frac12)\) \(\approx\) \(0.239920 - 1.01631i\)
\(L(2)\) 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 \( 1 + 3iT \)
5 \( 1 \)
good7 \( 1 + 2iT - 49T^{2} \)
11 \( 1 - 121T^{2} \)
13 \( 1 + 22iT - 169T^{2} \)
17 \( 1 + 289T^{2} \)
19 \( 1 + 26T + 361T^{2} \)
23 \( 1 + 529T^{2} \)
29 \( 1 - 841T^{2} \)
31 \( 1 + 46T + 961T^{2} \)
37 \( 1 + 26iT - 1.36e3T^{2} \)
41 \( 1 - 1.68e3T^{2} \)
43 \( 1 + 22iT - 1.84e3T^{2} \)
47 \( 1 + 2.20e3T^{2} \)
53 \( 1 + 2.80e3T^{2} \)
59 \( 1 - 3.48e3T^{2} \)
61 \( 1 - 74T + 3.72e3T^{2} \)
67 \( 1 + 122iT - 4.48e3T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 + 46iT - 5.32e3T^{2} \)
79 \( 1 - 142T + 6.24e3T^{2} \)
83 \( 1 + 6.88e3T^{2} \)
89 \( 1 - 7.92e3T^{2} \)
97 \( 1 + 2iT - 9.40e3T^{2} \)
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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

−11.05554593106336849182783927498, −10.44235121939280235955224392967, −9.017331377980182466971970882402, −8.084862373590894508443976560871, −7.32157117934016320868338868758, −6.21991501236242035894940905654, −5.27202927199347490164345987491, −3.56643886117874732401939577164, −2.18127135462061597737397001789, −0.48535084706982882982774333539, 2.21309151752648367900254771234, 3.80122920765454181374172985339, 4.66485083460652070047981802162, 5.88583030440727972690620875082, 6.94767676644916616494534931885, 8.469124244425378967815587703750, 9.106530733035266101682380655753, 9.997613870902732190259176458328, 11.02248816653733503534775540612, 11.65894457312569996036557439796

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