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Properties

Label 2-1950-65.49-c1-0-37
Degree $2$
Conductor $1950$
Sign $0.857 + 0.514i$
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 + (0.866 − 0.5i)3-s + (−0.499 + 0.866i)4-s + (0.866 + 0.499i)6-s + (2.36 − 4.09i)7-s − 0.999·8-s + (0.499 − 0.866i)9-s + (4.09 − 2.36i)11-s + 0.999i·12-s + (−0.232 + 3.59i)13-s + 4.73·14-s + (−0.5 − 0.866i)16-s + (−4.5 − 2.59i)17-s + 0.999·18-s + (−1.09 − 0.633i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.499 − 0.288i)3-s + (−0.249 + 0.433i)4-s + (0.353 + 0.204i)6-s + (0.894 − 1.54i)7-s − 0.353·8-s + (0.166 − 0.288i)9-s + (1.23 − 0.713i)11-s + 0.288i·12-s + (−0.0643 + 0.997i)13-s + 1.26·14-s + (−0.125 − 0.216i)16-s + (−1.09 − 0.630i)17-s + 0.235·18-s + (−0.251 − 0.145i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 + 0.514i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.857 + 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.857 + 0.514i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ 0.857 + 0.514i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.710853510\)
\(L(\frac12)\) \(\approx\) \(2.710853510\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 \)
13 \( 1 + (0.232 - 3.59i)T \)
good7 \( 1 + (-2.36 + 4.09i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-4.09 + 2.36i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (4.5 + 2.59i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.09 + 0.633i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.90 + 1.09i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.53iT - 31T^{2} \)
37 \( 1 + (-1.5 - 2.59i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.401 + 0.232i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.36 + 3.09i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 1.26T + 47T^{2} \)
53 \( 1 + 3iT - 53T^{2} \)
59 \( 1 + (-12 - 6.92i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.40 - 4.16i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.36 + 9.29i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-7.09 - 4.09i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 12.1T + 73T^{2} \)
79 \( 1 - 12.3T + 79T^{2} \)
83 \( 1 - 11.6T + 83T^{2} \)
89 \( 1 + (2.19 - 1.26i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (3 - 5.19i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.889223358292863701433306316237, −8.300248810143477997655133755913, −7.34447976168087620202038420773, −6.88810857115686142912643243470, −6.23378333044253775301368459393, −4.82196332536661924391250149447, −4.22372259785258019157549157549, −3.57795622736143595411142319256, −2.09123444306644730952363591276, −0.858711655237212459706073599104, 1.59886395187927515856132841985, 2.28823871036227476979611180151, 3.31109420990351141597572243431, 4.32502989527211776376102111295, 5.04419335629583856101811601557, 5.84618641188474730573467436870, 6.79293017691419969400257840782, 7.984721975031732710955075300379, 8.720117556730655748129813992262, 9.143614165922997894859984561040

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