Properties

Degree $2$
Conductor $1050$
Sign $-0.241 + 0.970i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + (−1.61 + 0.618i)3-s + 4-s + (1.61 − 0.618i)6-s + (2.61 + 0.381i)7-s − 8-s + (2.23 − 2.00i)9-s − 4.47i·11-s + (−1.61 + 0.618i)12-s − 3.23·13-s + (−2.61 − 0.381i)14-s + 16-s − 0.763i·17-s + (−2.23 + 2.00i)18-s − 0.472i·19-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.934 + 0.356i)3-s + 0.5·4-s + (0.660 − 0.252i)6-s + (0.989 + 0.144i)7-s − 0.353·8-s + (0.745 − 0.666i)9-s − 1.34i·11-s + (−0.467 + 0.178i)12-s − 0.897·13-s + (−0.699 − 0.102i)14-s + 0.250·16-s − 0.185i·17-s + (−0.527 + 0.471i)18-s − 0.108i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.241 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.241 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.241 + 0.970i$
Motivic weight: \(1\)
Character: $\chi_{1050} (1049, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ -0.241 + 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5182126142\)
\(L(\frac12)\) \(\approx\) \(0.5182126142\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (1.61 - 0.618i)T \)
5 \( 1 \)
7 \( 1 + (-2.61 - 0.381i)T \)
good11 \( 1 + 4.47iT - 11T^{2} \)
13 \( 1 + 3.23T + 13T^{2} \)
17 \( 1 + 0.763iT - 17T^{2} \)
19 \( 1 + 0.472iT - 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + 5.70iT - 29T^{2} \)
31 \( 1 - 7.23iT - 31T^{2} \)
37 \( 1 - 5.23iT - 37T^{2} \)
41 \( 1 + 6.47T + 41T^{2} \)
43 \( 1 + 12.9iT - 43T^{2} \)
47 \( 1 + 2.47iT - 47T^{2} \)
53 \( 1 + 8.47T + 53T^{2} \)
59 \( 1 + 4.47T + 59T^{2} \)
61 \( 1 + 2.76iT - 61T^{2} \)
67 \( 1 + 12iT - 67T^{2} \)
71 \( 1 + 2.76iT - 71T^{2} \)
73 \( 1 - 6.76T + 73T^{2} \)
79 \( 1 + 8.94T + 79T^{2} \)
83 \( 1 + 16.6iT - 83T^{2} \)
89 \( 1 + 14.4T + 89T^{2} \)
97 \( 1 - 5.23T + 97T^{2} \)
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

−9.826885127996695270448991933705, −8.831462342486234265809499776737, −8.133760482282560326374128253393, −7.20412132874865593758765174077, −6.25438630348587563263632508419, −5.42394924641096959119096631266, −4.62375780971466794182333738391, −3.31107826130635190812698025934, −1.78535800500740185151907143042, −0.34090529985331480129948075856, 1.42681952309950710163665546073, 2.29487039322055280608052430555, 4.28178024972992515984636792093, 5.00078550630960293214710899284, 5.99613106683636009633390000974, 7.06436220287970016330798350741, 7.55126189349917732143730036245, 8.293293513046668978587101769173, 9.626348137728311260328637809965, 10.05727843316326755738361798829

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