Properties

Label 2-1950-13.10-c1-0-8
Degree $2$
Conductor $1950$
Sign $0.454 - 0.890i$
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  + (0.866 − 0.5i)2-s + (−0.5 − 0.866i)3-s + (0.499 − 0.866i)4-s + (−0.866 − 0.499i)6-s + (−1.09 − 0.633i)7-s − 0.999i·8-s + (−0.499 + 0.866i)9-s + (−1.09 + 0.633i)11-s − 0.999·12-s + (−1.59 + 3.23i)13-s − 1.26·14-s + (−0.5 − 0.866i)16-s + (−2.59 + 4.5i)17-s + 0.999i·18-s + (−4.09 − 2.36i)19-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.288 − 0.499i)3-s + (0.249 − 0.433i)4-s + (−0.353 − 0.204i)6-s + (−0.415 − 0.239i)7-s − 0.353i·8-s + (−0.166 + 0.288i)9-s + (−0.331 + 0.191i)11-s − 0.288·12-s + (−0.443 + 0.896i)13-s − 0.338·14-s + (−0.125 − 0.216i)16-s + (−0.630 + 1.09i)17-s + 0.235i·18-s + (−0.940 − 0.542i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.454 - 0.890i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.454 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.454 - 0.890i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (751, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ 0.454 - 0.890i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.072208184\)
\(L(\frac12)\) \(\approx\) \(1.072208184\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 \)
13 \( 1 + (1.59 - 3.23i)T \)
good7 \( 1 + (1.09 + 0.633i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.09 - 0.633i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.59 - 4.5i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.09 + 2.36i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.09 - 7.09i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.5 - 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 9.46iT - 31T^{2} \)
37 \( 1 + (-2.59 + 1.5i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.59 + 3.23i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.09 + 3.63i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 4.73iT - 47T^{2} \)
53 \( 1 + 3T + 53T^{2} \)
59 \( 1 + (12 + 6.92i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.59 - 13.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.29 - 3.63i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.90 - 1.09i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 12.1iT - 73T^{2} \)
79 \( 1 - 8.39T + 79T^{2} \)
83 \( 1 + 5.66iT - 83T^{2} \)
89 \( 1 + (8.19 - 4.73i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.19 - 3i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.289591186879934149050488216556, −8.672871030372100136263368464297, −7.44866187486580011610756932651, −6.86149597551814615564020294434, −6.20201852156915303392570568172, −5.22013205456007714625647560862, −4.45002159499126000701755496321, −3.49342218613693284100468809283, −2.38877505130440582184154181362, −1.43568392168370090733788668972, 0.31413193449741387225835876737, 2.50768481078637230066765402810, 3.10424065668116919733448059319, 4.48375720134840243426692458265, 4.76460831320810299779223629097, 6.09045037045612896269046654805, 6.21979453775535520421196817304, 7.49334371567726320027053415913, 8.112451306388805840476045222276, 9.134448682433627299357870547619

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