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Properties

Label 2-1950-65.4-c1-0-4
Degree $2$
Conductor $1950$
Sign $-0.804 + 0.593i$
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 + (2.56 + 4.44i)7-s + 0.999·8-s + (0.499 + 0.866i)9-s + (−4.83 − 2.78i)11-s − 0.999i·12-s + (−1.67 + 3.19i)13-s − 5.13·14-s + (−0.5 + 0.866i)16-s + (−1.11 + 0.641i)17-s − 0.999·18-s + (−5.75 + 3.32i)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 + (0.969 + 1.67i)7-s + 0.353·8-s + (0.166 + 0.288i)9-s + (−1.45 − 0.841i)11-s − 0.288i·12-s + (−0.465 + 0.884i)13-s − 1.37·14-s + (−0.125 + 0.216i)16-s + (−0.269 + 0.155i)17-s − 0.235·18-s + (−1.32 + 0.762i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.804 + 0.593i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.804 + 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-0.804 + 0.593i$
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: \((2,\ 1950,\ (\ :1/2),\ -0.804 + 0.593i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6624664717\)
\(L(\frac12)\) \(\approx\) \(0.6624664717\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.67 - 3.19i)T \)
good7 \( 1 + (-2.56 - 4.44i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (4.83 + 2.78i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.11 - 0.641i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.75 - 3.32i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.94 + 3.43i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.75 + 3.03i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.30iT - 31T^{2} \)
37 \( 1 + (-4.19 + 7.25i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.11 - 0.641i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.77 - 2.17i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 5.02T + 47T^{2} \)
53 \( 1 - 6.30iT - 53T^{2} \)
59 \( 1 + (1.31 - 0.759i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.19 - 8.99i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.43 + 2.48i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-9.44 + 5.45i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 4.94T + 73T^{2} \)
79 \( 1 + 10.1T + 79T^{2} \)
83 \( 1 - 17.3T + 83T^{2} \)
89 \( 1 + (5.57 + 3.22i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.58 - 6.20i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.377282823386561663190911280978, −8.696017504524458529175596263822, −8.102718118437853322408221757452, −7.79854677160860952188096469637, −6.26121565062784298120147404819, −5.81369305532966922651379991484, −4.90343036096147168079233760225, −4.13837079280772576260927997182, −2.46225890392534059914755603731, −2.11499000083912831545457710924, 0.23193307691663124022822602636, 1.59226242028706743300046215697, 2.48968344067262804277056280917, 3.56494258128782683913697308352, 4.59406636053331841015496977856, 5.06334133801874507458610719484, 6.75844792498507699423392146734, 7.37849629950380136716356523107, 8.093735387283926877271976828501, 8.391588612604406729970534513954

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