Properties

Label 2-1050-7.2-c1-0-14
Degree $2$
Conductor $1050$
Sign $0.605 - 0.795i$
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  + (−0.5 + 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s − 0.999·6-s + (2.5 − 0.866i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (1 + 1.73i)11-s + (0.499 − 0.866i)12-s + 4·13-s + (−0.500 + 2.59i)14-s + (−0.5 + 0.866i)16-s + (−2.5 − 4.33i)17-s + (−0.499 − 0.866i)18-s + (2 − 3.46i)19-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 + (0.944 − 0.327i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (0.301 + 0.522i)11-s + (0.144 − 0.249i)12-s + 1.10·13-s + (−0.133 + 0.694i)14-s + (−0.125 + 0.216i)16-s + (−0.606 − 1.05i)17-s + (−0.117 − 0.204i)18-s + (0.458 − 0.794i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.605 - 0.795i$
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: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ 0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.709356042\)
\(L(\frac12)\) \(\approx\) \(1.709356042\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 + 0.866i)T \)
good11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 + (2.5 + 4.33i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2 + 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.5 + 4.33i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + (-5.5 - 9.52i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4 + 6.92i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 5T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + (0.5 - 0.866i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6 - 10.3i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1 - 1.73i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5 - 8.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + T + 71T^{2} \)
73 \( 1 + (-1 - 1.73i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.5 - 7.79i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (5.5 - 9.52i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - T + 97T^{2} \)
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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

−9.866432794298901754021293097687, −8.950798334090935621630052386204, −8.614171111667303945006922285519, −7.46866954005042413085886940372, −6.93254852802980463489676724458, −5.72578682049746205907331564486, −4.76534720584507858946737659430, −4.14838517495850599101002565280, −2.65851485580606563153768785084, −1.13310188014469602296077227228, 1.18800209187958981410913311556, 2.05653696486711078727303000129, 3.38930087982943395120285233898, 4.22928170438836091623492441856, 5.61356123155920467164289765234, 6.36365353699598227377158881534, 7.70128489669497258015131771057, 8.190847498103796296484384017440, 8.883064094953710124170308552058, 9.701070766085059643832054411710

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