Properties

Label 2-1050-5.4-c1-0-14
Degree $2$
Conductor $1050$
Sign $-0.894 + 0.447i$
Analytic cond. $8.38429$
Root an. cond. $2.89556$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  i·2-s + i·3-s − 4-s + 6-s + i·7-s + i·8-s − 9-s − 4·11-s i·12-s − 2i·13-s + 14-s + 16-s − 2i·17-s + i·18-s − 4·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.408·6-s + 0.377i·7-s + 0.353i·8-s − 0.333·9-s − 1.20·11-s − 0.288i·12-s − 0.554i·13-s + 0.267·14-s + 0.250·16-s − 0.485i·17-s + 0.235i·18-s − 0.917·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(8.38429\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (799, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5450538133\)
\(L(\frac12)\) \(\approx\) \(0.5450538133\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 - iT \)
5 \( 1 \)
7 \( 1 - iT \)
good11 \( 1 + 4T + 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + 8iT - 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 6iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 10iT - 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 - 14T + 61T^{2} \)
67 \( 1 - 12iT - 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 - 10iT - 73T^{2} \)
79 \( 1 + 16T + 79T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 + 2iT - 97T^{2} \)
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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

−9.871803531896670895488597185168, −8.639645902865867538134333736917, −8.370954911433617554404760448593, −7.09861969659027575895413849968, −5.85525530889552964967022684095, −5.06175895540358495488219928105, −4.24045980674624965924177980555, −3.01078274682895516553064085863, −2.27374402152380305517877193771, −0.23642560944077347037438849255, 1.60551894362912374085680735087, 3.04483606086810999659184355766, 4.28934501251681630997564783480, 5.26263479452017392656278536065, 6.15420714896632392776529592794, 6.95129206243967525854584535116, 7.76958854243572698548234169468, 8.311706994511949254366048060545, 9.298537464081603216212391497777, 10.20977089497244142164150412540

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