↑

Properties

Label 2-1960-7.4-c1-0-6
Degree $2$
Conductor $1960$
Sign $-0.605 - 0.795i$
Analytic cond. $15.6506$
Root an. cond. $3.95609$
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  + (0.207 − 0.358i)3-s + (0.5 + 0.866i)5-s + (1.41 + 2.44i)9-s + (−0.414 + 0.717i)11-s − 2·13-s + 0.414·15-s + (−3.82 + 6.63i)17-s + (−2.82 − 4.89i)19-s + (2.79 + 4.83i)23-s + (−0.499 + 0.866i)25-s + 2.41·27-s − 7.82·29-s + (0.414 − 0.717i)31-s + (0.171 + 0.297i)33-s + (−2.82 − 4.89i)37-s + ⋯
L(s)  = 1  + (0.119 − 0.207i)3-s + (0.223 + 0.387i)5-s + (0.471 + 0.816i)9-s + (−0.124 + 0.216i)11-s − 0.554·13-s + 0.106·15-s + (−0.928 + 1.60i)17-s + (−0.648 − 1.12i)19-s + (0.582 + 1.00i)23-s + (−0.0999 + 0.173i)25-s + 0.464·27-s − 1.45·29-s + (0.0743 − 0.128i)31-s + (0.0298 + 0.0517i)33-s + (−0.464 − 0.805i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1960\)    =    \(2^{3} \cdot 5 \cdot 7^{2}\)
Sign: $-0.605 - 0.795i$
Analytic conductor: \(15.6506\)
Root analytic conductor: \(3.95609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1960} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1960,\ (\ :1/2),\ -0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.061540271\)
\(L(\frac12)\) \(\approx\) \(1.061540271\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 \)
good3 \( 1 + (-0.207 + 0.358i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (0.414 - 0.717i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 + (3.82 - 6.63i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.82 + 4.89i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.79 - 4.83i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 7.82T + 29T^{2} \)
31 \( 1 + (-0.414 + 0.717i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.82 + 4.89i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 5.82T + 41T^{2} \)
43 \( 1 + 6.89T + 43T^{2} \)
47 \( 1 + (-5.82 - 10.0i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.82 + 4.89i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2 - 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.32 - 5.76i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.44 + 11.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + (-1.82 + 3.16i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 4.75T + 83T^{2} \)
89 \( 1 + (2.67 + 4.62i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 6T + 97T^{2} \)
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

−9.388389018907434691820870710374, −8.712218299286858589360286919674, −7.76491532195632029919152256042, −7.15539590790976928693264785975, −6.43248918749161162828341245657, −5.42256140193026794703822297835, −4.60161804479991093767367404680, −3.66005570370460033093663051106, −2.39045374121638507053395795534, −1.73179142722232328469696462737, 0.34941546957558404290195694030, 1.82965507482235433738511637111, 2.95792377415495318627261965315, 3.99850507132720828686530795474, 4.80683897202288295978781444565, 5.60101351024758720380747272725, 6.71685422101951450551063410343, 7.13382872651424271154826443545, 8.361789875835579286267492071705, 8.893147859289442013440607487416

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