↑

Properties

Label 2-300-20.7-c1-0-2
Degree $2$
Conductor $300$
Sign $-0.189 - 0.981i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 − 0.707i)2-s + (0.707 + 0.707i)3-s + (0.999 + 1.73i)4-s + (−0.366 − 1.36i)6-s + (−3.15 + 3.15i)7-s − 2.82i·8-s + 1.00i·9-s + 2i·11-s + (−0.517 + 1.93i)12-s + (−1.60 + 1.60i)13-s + (6.09 − 1.63i)14-s + (−2.00 + 3.46i)16-s + (−4.24 − 4.24i)17-s + (0.707 − 1.22i)18-s − 3.19·19-s + ⋯
L(s)  = 1  + (−0.866 − 0.499i)2-s + (0.408 + 0.408i)3-s + (0.499 + 0.866i)4-s + (−0.149 − 0.557i)6-s + (−1.19 + 1.19i)7-s − 0.999i·8-s + 0.333i·9-s + 0.603i·11-s + (−0.149 + 0.557i)12-s + (−0.444 + 0.444i)13-s + (1.62 − 0.436i)14-s + (−0.500 + 0.866i)16-s + (−1.02 − 1.02i)17-s + (0.166 − 0.288i)18-s − 0.733·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.189 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.189 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.189 - 0.981i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ -0.189 - 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.410778 + 0.497861i\)
\(L(\frac12)\) \(\approx\) \(0.410778 + 0.497861i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 + 0.707i)T \)
3 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 \)
good7 \( 1 + (3.15 - 3.15i)T - 7iT^{2} \)
11 \( 1 - 2iT - 11T^{2} \)
13 \( 1 + (1.60 - 1.60i)T - 13iT^{2} \)
17 \( 1 + (4.24 + 4.24i)T + 17iT^{2} \)
19 \( 1 + 3.19T + 19T^{2} \)
23 \( 1 + (-5.27 - 5.27i)T + 23iT^{2} \)
29 \( 1 - 0.535iT - 29T^{2} \)
31 \( 1 - 3.73iT - 31T^{2} \)
37 \( 1 + (-7.72 - 7.72i)T + 37iT^{2} \)
41 \( 1 - 1.46T + 41T^{2} \)
43 \( 1 + (-3.15 - 3.15i)T + 43iT^{2} \)
47 \( 1 + (-0.656 + 0.656i)T - 47iT^{2} \)
53 \( 1 + (-3.86 + 3.86i)T - 53iT^{2} \)
59 \( 1 + 11.4T + 59T^{2} \)
61 \( 1 - 3T + 61T^{2} \)
67 \( 1 + (-3.91 + 3.91i)T - 67iT^{2} \)
71 \( 1 + 2.53iT - 71T^{2} \)
73 \( 1 + (-2.82 + 2.82i)T - 73iT^{2} \)
79 \( 1 - 7.46T + 79T^{2} \)
83 \( 1 + (9.14 + 9.14i)T + 83iT^{2} \)
89 \( 1 - 6.92iT - 89T^{2} \)
97 \( 1 + (-6.88 - 6.88i)T + 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.88327506972296212069204971236, −10.99735392260864632992072506330, −9.725667798768387621640593350921, −9.369802173269049747610101373374, −8.636197028246861429443753091537, −7.29186820046988498739396883288, −6.39099051004014484304902242892, −4.71745284857525853345750857197, −3.17416933566378841502331827245, −2.29727584641185194499338909497, 0.56835596978907116129257804668, 2.58457900479375986576438811531, 4.15779414298777680929023556903, 6.03208701118466214981345176209, 6.73976577893758653843966181014, 7.59761225680451733627028966123, 8.614250589012497145264576761881, 9.447486337423777875147635587304, 10.49958670209879779078600004054, 10.99391320717495236256663147231

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