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Properties

Label 2-1950-65.4-c1-0-12
Degree $2$
Conductor $1950$
Sign $-0.658 - 0.752i$
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 + (1.5 + 2.59i)7-s + 0.999·8-s + (0.499 + 0.866i)9-s + (−3.23 − 1.86i)11-s − 0.999i·12-s + (3.5 − 0.866i)13-s − 3·14-s + (−0.5 + 0.866i)16-s + (−3.46 + 2i)17-s − 0.999·18-s + (1.96 − 1.13i)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 + (0.566 + 0.981i)7-s + 0.353·8-s + (0.166 + 0.288i)9-s + (−0.974 − 0.562i)11-s − 0.288i·12-s + (0.970 − 0.240i)13-s − 0.801·14-s + (−0.125 + 0.216i)16-s + (−0.840 + 0.485i)17-s − 0.235·18-s + (0.450 − 0.260i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.658 - 0.752i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.658 - 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-0.658 - 0.752i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ -0.658 - 0.752i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.540832993\)
\(L(\frac12)\) \(\approx\) \(1.540832993\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.5 + 0.866i)T \)
good7 \( 1 + (-1.5 - 2.59i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (3.23 + 1.86i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (3.46 - 2i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.96 + 1.13i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3 - 1.73i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.73 - 4.73i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 8.92iT - 31T^{2} \)
37 \( 1 + (3.96 - 6.86i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.46 - 2i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.19 + 3i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 0.464T + 47T^{2} \)
53 \( 1 + 3.73iT - 53T^{2} \)
59 \( 1 + (3.92 - 2.26i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.73 - 6.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.73 + 4.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.803 - 0.464i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 6.92T + 73T^{2} \)
79 \( 1 + 16.9T + 79T^{2} \)
83 \( 1 - 2.53T + 83T^{2} \)
89 \( 1 + (-8.76 - 5.06i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.19 + 10.7i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.965203560268945126103803145183, −8.773865695739033988498052684977, −8.129776085670058676094986918650, −7.26647466909064490073694056063, −6.30081815158833371619385505464, −5.38497579962527784224650990777, −4.93686525618766859552425449750, −3.59369203730034410161808166629, −2.65513609387938025199748331114, −1.41800325894065019969365434806, 0.61284921550639710534244728043, 1.84641394623897468406935452787, 2.70281238004129357633447137931, 3.92272466743445211565200754301, 4.45968776680412345934241876520, 5.64588348886244351261702141529, 6.84743438105733049375936920672, 7.61626730548800184274230258722, 8.009162545544668729530922823258, 9.013747929117674880822295631300

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