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Properties

Label 2-1050-35.9-c1-0-3
Degree $2$
Conductor $1050$
Sign $-0.441 - 0.897i$
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  + (−0.866 − 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s + 0.999·6-s + (0.866 + 2.5i)7-s − 0.999i·8-s + (0.499 − 0.866i)9-s + (2 + 3.46i)11-s + (−0.866 − 0.499i)12-s + 4i·13-s + (0.500 − 2.59i)14-s + (−0.5 + 0.866i)16-s + (−2.59 + 1.5i)17-s + (−0.866 + 0.499i)18-s + (3 − 5.19i)19-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.499 + 0.288i)3-s + (0.249 + 0.433i)4-s + 0.408·6-s + (0.327 + 0.944i)7-s − 0.353i·8-s + (0.166 − 0.288i)9-s + (0.603 + 1.04i)11-s + (−0.249 − 0.144i)12-s + 1.10i·13-s + (0.133 − 0.694i)14-s + (−0.125 + 0.216i)16-s + (−0.630 + 0.363i)17-s + (−0.204 + 0.117i)18-s + (0.688 − 1.19i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.441 - 0.897i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.441 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.441 - 0.897i$
Analytic conductor: \(8.38429\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ -0.441 - 0.897i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7474445962\)
\(L(\frac12)\) \(\approx\) \(0.7474445962\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 \)
7 \( 1 + (-0.866 - 2.5i)T \)
good11 \( 1 + (-2 - 3.46i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 4iT - 13T^{2} \)
17 \( 1 + (2.59 - 1.5i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3 + 5.19i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (6.06 + 3.5i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 + (-2.5 - 4.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.73 - i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 7T + 41T^{2} \)
43 \( 1 - 2iT - 43T^{2} \)
47 \( 1 + (-0.866 - 0.5i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.73 - i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (7 + 12.1i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6 - 10.3i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (10.3 - 6i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 9T + 71T^{2} \)
73 \( 1 + (-5.19 + 3i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (8.5 - 14.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 4iT - 83T^{2} \)
89 \( 1 + (3.5 - 6.06i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 7iT - 97T^{2} \)
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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

−10.09357426417801969091596716829, −9.235497117752406547038399096655, −8.945069155461368679403326536962, −7.75384730984233967347302589674, −6.80168454967231971146003665678, −6.09368405199112101606610269906, −4.79779657234692362695563872480, −4.15390117685510508511395262588, −2.58629654015027083714108278360, −1.60615892389035117706745638350, 0.45456962499879426680022132415, 1.60608828062511954445824033324, 3.33553854581546263619774924671, 4.43621677684662993206411573239, 5.81094038181987235180791179757, 6.06897577110880038463292805474, 7.51061097012305504555772566713, 7.67796646940637326345590715811, 8.712132607724647485274520516549, 9.752377025845569760952657515088

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