Properties

Label 2-1050-7.2-c1-0-24
Degree $2$
Conductor $1050$
Sign $-0.386 - 0.922i$
Analytic cond. $8.38429$
Root an. cond. $2.89556$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s − 0.999·6-s + (−0.5 + 2.59i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (−2.5 − 4.33i)11-s + (−0.499 + 0.866i)12-s + (2 + 1.73i)14-s + (−0.5 + 0.866i)16-s + (−2 − 3.46i)17-s + (0.499 + 0.866i)18-s + (−4 + 6.92i)19-s + (2.5 − 0.866i)21-s − 5·22-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s − 0.408·6-s + (−0.188 + 0.981i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.753 − 1.30i)11-s + (−0.144 + 0.249i)12-s + (0.534 + 0.462i)14-s + (−0.125 + 0.216i)16-s + (−0.485 − 0.840i)17-s + (0.117 + 0.204i)18-s + (−0.917 + 1.58i)19-s + (0.545 − 0.188i)21-s − 1.06·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.386 - 0.922i$
Analytic conductor: \(8.38429\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (751, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ -0.386 - 0.922i)\)

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 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 \)
7 \( 1 + (0.5 - 2.59i)T \)
good11 \( 1 + (2.5 + 4.33i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (2 + 3.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4 - 6.92i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 5T + 29T^{2} \)
31 \( 1 + (1.5 + 2.59i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2 - 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 2T + 43T^{2} \)
47 \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.5 + 7.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.5 - 9.52i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3 + 5.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1 + 1.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + (-5 - 8.66i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.5 - 2.59i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 7T + 83T^{2} \)
89 \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 7T + 97T^{2} \)
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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

−9.434956655005391004339348037327, −8.493348126849116086106028590215, −7.86893710961468314708691687304, −6.51360396989113378177973354703, −5.76410473914077211476771835103, −5.21668572551171480318499492699, −3.78681309717082734740436612075, −2.78962558431298810407728049052, −1.78347108183832061810664122247, 0, 2.25711565454085477630407995337, 3.73006254594142138554368954655, 4.50595392829906558172204785000, 5.13071125691891994709698534688, 6.41007476745326135587539873599, 6.97977074030033089473429676998, 7.82401896633040635985045540793, 8.810438674498531201628486466349, 9.678689101341295452822122358106

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