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Properties

Label 2-1950-65.4-c1-0-13
Degree $2$
Conductor $1950$
Sign $0.206 - 0.978i$
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 + (0.232 + 0.133i)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 + (−4.96 + 2.86i)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 + (0.0699 + 0.0403i)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 + (−1.13 + 0.657i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.206 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.206 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.206 - 0.978i$
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: \((2,\ 1950,\ (\ :1/2),\ 0.206 - 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.270829726\)
\(L(\frac12)\) \(\approx\) \(1.270829726\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.232 - 0.133i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-3.46 + 2i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.96 - 2.86i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3 - 1.73i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.732 + 1.26i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4.92iT - 31T^{2} \)
37 \( 1 + (-2.96 + 5.13i)T + (-18.5 - 32.0i)T^{2} \)
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 \( 1 - 6.46T + 47T^{2} \)
53 \( 1 - 0.267iT - 53T^{2} \)
59 \( 1 + (-9.92 + 5.73i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.267 - 0.464i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.732 - 1.26i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (11.1 - 6.46i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 6.92T + 73T^{2} \)
79 \( 1 + 3.07T + 79T^{2} \)
83 \( 1 - 9.46T + 83T^{2} \)
89 \( 1 + (-12.2 - 7.06i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.19 - 7.26i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.079671160023642177099906488504, −8.519698527619795107517255784089, −7.85928740090406698163979071008, −6.97680977569169646796853536017, −6.10982331571858888452229545531, −5.55901273552213636163081027197, −4.82176699759887838096354829144, −3.60301788456713829409251235302, −2.19344963941081354114626516443, −1.08076511406329344796919726155, 0.70118150265477416887726142717, 1.70999028271412598695349522804, 3.15990392642374962054066467665, 4.12490396768012320428632664951, 4.64255641108054443677641647082, 5.82089796822931547569123152291, 6.69671762119779998915690477180, 7.53205375876551158688981311365, 8.433641941386513725511761573867, 8.978986090467798706395430859642

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