↑

Properties

Label 2-780-13.9-c1-0-9
Degree $2$
Conductor $780$
Sign $-0.872 - 0.488i$
Analytic cond. $6.22833$
Root an. cond. $2.49566$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)3-s − 5-s + (−1.5 − 2.59i)7-s + (−0.499 − 0.866i)9-s + (−3 + 5.19i)11-s + (−2.5 + 2.59i)13-s + (−0.5 + 0.866i)15-s + (−3 − 5.19i)17-s − 3·21-s + (−1 + 1.73i)23-s + 25-s − 0.999·27-s + (−4 + 6.92i)29-s + 3·31-s + (3 + 5.19i)33-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s − 0.447·5-s + (−0.566 − 0.981i)7-s + (−0.166 − 0.288i)9-s + (−0.904 + 1.56i)11-s + (−0.693 + 0.720i)13-s + (−0.129 + 0.223i)15-s + (−0.727 − 1.26i)17-s − 0.654·21-s + (−0.208 + 0.361i)23-s + 0.200·25-s − 0.192·27-s + (−0.742 + 1.28i)29-s + 0.538·31-s + (0.522 + 0.904i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 - 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.872 - 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(780\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.872 - 0.488i$
Analytic conductor: \(6.22833\)
Root analytic conductor: \(2.49566\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{780} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 780,\ (\ :1/2),\ -0.872 - 0.488i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + T \)
13 \( 1 + (2.5 - 2.59i)T \)
good7 \( 1 + (1.5 + 2.59i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (3 - 5.19i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (3 + 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1 - 1.73i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4 - 6.92i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 3T + 31T^{2} \)
37 \( 1 + (3 - 5.19i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1 - 1.73i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.5 + 9.52i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 12T + 47T^{2} \)
53 \( 1 + 12T + 53T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.5 + 12.9i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.5 + 4.33i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4 - 6.92i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 5T + 73T^{2} \)
79 \( 1 + 15T + 79T^{2} \)
83 \( 1 + 8T + 83T^{2} \)
89 \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T + (-48.5 + 84.0i)T^{2} \)
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

−9.795253930744951699615387550641, −9.054686324219062133695924526068, −7.81002722897729637007986197791, −7.08354492990191900589221686181, −6.86052594152187956283515096527, −5.10043143192897761373195461623, −4.34606384107066402717759143104, −3.10400929066813954291693373265, −1.91960735220724189863127819868, 0, 2.48850257529594289356702580187, 3.24634243375825799780929306965, 4.38605024292144570781445609224, 5.64949132866934028433229081683, 6.10535413555741795213766257674, 7.63210577271260567895863451874, 8.377989381547274466090696391825, 8.937031556415677912721709545305, 10.00128469514225780138232708133

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