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Properties

Label 2-1950-13.10-c1-0-43
Degree $2$
Conductor $1950$
Sign $-0.505 - 0.862i$
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 + (−2.53 − 1.46i)7-s − 0.999i·8-s + (−0.499 + 0.866i)9-s + (−0.992 + 0.573i)11-s − 0.999·12-s + (−1.86 − 3.08i)13-s − 2.93·14-s + (−0.5 − 0.866i)16-s + (−0.276 + 0.478i)17-s + 0.999i·18-s + (0.723 + 0.417i)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 + (−0.959 − 0.554i)7-s − 0.353i·8-s + (−0.166 + 0.288i)9-s + (−0.299 + 0.172i)11-s − 0.288·12-s + (−0.516 − 0.856i)13-s − 0.783·14-s + (−0.125 − 0.216i)16-s + (−0.0670 + 0.116i)17-s + 0.235i·18-s + (0.165 + 0.0958i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.505 - 0.862i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.505 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-0.505 - 0.862i$
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: \((2,\ 1950,\ (\ :1/2),\ -0.505 - 0.862i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2606712730\)
\(L(\frac12)\) \(\approx\) \(0.2606712730\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.86 + 3.08i)T \)
good7 \( 1 + (2.53 + 1.46i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.992 - 0.573i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.276 - 0.478i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.723 - 0.417i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.496 + 0.859i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.03 - 1.79i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.36iT - 31T^{2} \)
37 \( 1 + (7.00 - 4.04i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (9.67 - 5.58i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.94 + 6.82i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 10.9iT - 47T^{2} \)
53 \( 1 + 3.56T + 53T^{2} \)
59 \( 1 + (4.75 + 2.74i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.43 - 2.48i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (8.88 - 5.13i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-9.85 - 5.69i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 8.19iT - 73T^{2} \)
79 \( 1 - 1.68T + 79T^{2} \)
83 \( 1 + 8.77iT - 83T^{2} \)
89 \( 1 + (3.93 - 2.26i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (9.19 + 5.30i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.632817327415054494607267865764, −7.66845752955919216169719396043, −6.93932904417185883848125000660, −6.30051685755310505996470616633, −5.38683872142109906746243272745, −4.64557976361585748804222455938, −3.44120058700279068928169196203, −2.81161887423024678022611472972, −1.46928596419998840374146183691, −0.07504196675700732889825684423, 2.16868600403276397890902007575, 3.17984138237762178118503803310, 4.00182409486175123988698492817, 4.96690344822942118232556324511, 5.65353364448617365062612673882, 6.46297286131981035319025702239, 7.08092004059855159897240535230, 8.110621417526964347688335437728, 9.069564597564172041278298739960, 9.581078295469808683588944141656

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