Properties

Label 2-1950-65.29-c1-0-28
Degree $2$
Conductor $1950$
Sign $0.540 + 0.841i$
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 + (0.866 − 0.5i)3-s + (0.499 − 0.866i)4-s + (0.499 − 0.866i)6-s + (2.59 + 1.5i)7-s − 0.999i·8-s + (0.499 − 0.866i)9-s + (−0.5 − 0.866i)11-s − 0.999i·12-s + (2.59 − 2.5i)13-s + 3·14-s + (−0.5 − 0.866i)16-s − 0.999i·18-s + (−2.5 + 4.33i)19-s + 3·21-s + (−0.866 − 0.499i)22-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.499 − 0.288i)3-s + (0.249 − 0.433i)4-s + (0.204 − 0.353i)6-s + (0.981 + 0.566i)7-s − 0.353i·8-s + (0.166 − 0.288i)9-s + (−0.150 − 0.261i)11-s − 0.288i·12-s + (0.720 − 0.693i)13-s + 0.801·14-s + (−0.125 − 0.216i)16-s − 0.235i·18-s + (−0.573 + 0.993i)19-s + 0.654·21-s + (−0.184 − 0.106i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.540 + 0.841i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.540 + 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.540 + 0.841i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (1849, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ 0.540 + 0.841i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.490647573\)
\(L(\frac12)\) \(\approx\) \(3.490647573\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 \)
13 \( 1 + (-2.59 + 2.5i)T \)
good7 \( 1 + (-2.59 - 1.5i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.5 - 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.46 + 2i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 10T + 31T^{2} \)
37 \( 1 + (0.866 - 0.5i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (3 + 5.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.73 + i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 9iT - 47T^{2} \)
53 \( 1 - 13iT - 53T^{2} \)
59 \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (10.3 - 6i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1 + 1.73i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 16iT - 73T^{2} \)
79 \( 1 - 10T + 79T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (10.3 + 6i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.722977334353497959341185837325, −8.453760671093090757601273565914, −7.60931145085447935601592650276, −6.54173596906625829547403904917, −5.76738666415937700560585516389, −4.98649936128342067461450213170, −4.03341269987190224942214397921, −3.07827941879822769644181305241, −2.18209457349001252824412030441, −1.13994476464709636524394066693, 1.41013736048468723594731439097, 2.59444830673061003757048273862, 3.63479778008292596754054398793, 4.63211044674775159380957326690, 4.85568446522594580590303025835, 6.22787042431572163422041336642, 6.91744708108375271935424553366, 7.78591773425927211742643925153, 8.391468159437167466493013009523, 9.135102954329212783389807679659

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