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Properties

Label 2-1050-105.104-c1-0-26
Degree $2$
Conductor $1050$
Sign $0.651 - 0.758i$
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  + 2-s + (1.22 + 1.22i)3-s + 4-s + (1.22 + 1.22i)6-s + (2.44 + i)7-s + 8-s + 2.99i·9-s + (1.22 + 1.22i)12-s + 2.44·13-s + (2.44 + i)14-s + 16-s − 4.89i·17-s + 2.99i·18-s − 2.44i·19-s + (1.77 + 4.22i)21-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.707 + 0.707i)3-s + 0.5·4-s + (0.499 + 0.499i)6-s + (0.925 + 0.377i)7-s + 0.353·8-s + 0.999i·9-s + (0.353 + 0.353i)12-s + 0.679·13-s + (0.654 + 0.267i)14-s + 0.250·16-s − 1.18i·17-s + 0.707i·18-s − 0.561i·19-s + (0.387 + 0.921i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.651 - 0.758i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.651 - 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.651 - 0.758i$
Analytic conductor: \(8.38429\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (1049, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ 0.651 - 0.758i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.468193258\)
\(L(\frac12)\) \(\approx\) \(3.468193258\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.22 - 1.22i)T \)
5 \( 1 \)
7 \( 1 + (-2.44 - i)T \)
good11 \( 1 - 11T^{2} \)
13 \( 1 - 2.44T + 13T^{2} \)
17 \( 1 + 4.89iT - 17T^{2} \)
19 \( 1 + 2.44iT - 19T^{2} \)
23 \( 1 + 6T + 23T^{2} \)
29 \( 1 - 6iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + 4.89T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 + 4.89iT - 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 - 12.2T + 59T^{2} \)
61 \( 1 + 12.2iT - 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + 9.79T + 73T^{2} \)
79 \( 1 - 10T + 79T^{2} \)
83 \( 1 - 2.44iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 4.89T + 97T^{2} \)
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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

−10.04885671352957808973073726351, −9.154184087851740065877350617786, −8.370129100131954520347343539869, −7.66669743442012623598323928393, −6.59195397511434724454294308115, −5.34440495652606463811328999954, −4.83383220856005023640109665178, −3.83986375980348119262798929270, −2.86739255460033255843117141216, −1.80267881192866430480556060680, 1.38788309475886548449170190107, 2.27452239723119993113642430341, 3.69425169405336512398475280866, 4.22092591541903285299094398198, 5.66557650030998166521366179812, 6.32784738049015520781449809598, 7.36750031452732742085532120594, 8.109468528279332555153955980355, 8.590944948917136219772609480436, 9.909680715209230952096241611298

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