Properties

Label 2-1950-13.9-c1-0-24
Degree $2$
Conductor $1950$
Sign $0.686 + 0.727i$
Analytic cond. $15.5708$
Root an. cond. $3.94598$
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 + (0.499 + 0.866i)6-s + (1.58 + 2.73i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (−1.5 + 2.59i)11-s − 0.999·12-s + (−3.08 − 1.87i)13-s − 3.16·14-s + (−0.5 + 0.866i)16-s + (−3.58 − 6.20i)17-s + 0.999·18-s + (−1.58 − 2.73i)19-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.204 + 0.353i)6-s + (0.597 + 1.03i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.452 + 0.783i)11-s − 0.288·12-s + (−0.854 − 0.519i)13-s − 0.845·14-s + (−0.125 + 0.216i)16-s + (−0.868 − 1.50i)17-s + 0.235·18-s + (−0.362 − 0.628i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.686 + 0.727i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.686 + 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.686 + 0.727i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ 0.686 + 0.727i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.237254215\)
\(L(\frac12)\) \(\approx\) \(1.237254215\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 + (3.08 + 1.87i)T \)
good7 \( 1 + (-1.58 - 2.73i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (3.58 + 6.20i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.58 + 2.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.08 + 3.60i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.16 + 7.20i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 9.16T + 31T^{2} \)
37 \( 1 + (5.24 - 9.08i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.74 + 8.21i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.58 + 4.47i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 - 7.16T + 53T^{2} \)
59 \( 1 + (-3 - 5.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.24 + 12.5i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2 - 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.91 - 6.78i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 1.32T + 73T^{2} \)
79 \( 1 - 9.16T + 79T^{2} \)
83 \( 1 + 13.6T + 83T^{2} \)
89 \( 1 + (-3.58 + 6.20i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.33 + 4.04i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.721783693375064207426898913644, −8.464565421059372089002965424747, −7.42419593541427903735571781690, −6.96384126277614031612177453373, −6.03125250688740909450100964159, −4.95598383777951007707517980665, −4.64871640336675373919830711312, −2.63053037129713109333971092588, −2.30632685065632745382054816566, −0.52381295905200796235195739110, 1.20454305175708519780431294111, 2.35334869705212291475025570743, 3.45963764086113994637062361783, 4.25149522225860850612567155033, 4.90114992106952499241106154258, 6.11888003150890001617134066072, 7.17160685677043275428771001640, 7.963401398854650979595282409162, 8.569025222027595768025922281372, 9.281376657169004407012582717455

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