Properties

Label 2-1950-13.10-c1-0-17
Degree $2$
Conductor $1950$
Sign $0.946 + 0.322i$
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 + (−2.32 − 1.34i)7-s + 0.999i·8-s + (−0.499 + 0.866i)9-s + (4.61 − 2.66i)11-s − 0.999·12-s + (3.42 + 1.12i)13-s + 2.68·14-s + (−0.5 − 0.866i)16-s + (−2.18 + 3.78i)17-s − 0.999i·18-s + (2.70 + 1.56i)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 + (−0.880 − 0.508i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (1.39 − 0.802i)11-s − 0.288·12-s + (0.950 + 0.310i)13-s + 0.718·14-s + (−0.125 − 0.216i)16-s + (−0.529 + 0.917i)17-s − 0.235i·18-s + (0.620 + 0.358i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 + 0.322i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.946 + 0.322i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.946 + 0.322i$
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: \((2,\ 1950,\ (\ :1/2),\ 0.946 + 0.322i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.120500001\)
\(L(\frac12)\) \(\approx\) \(1.120500001\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.42 - 1.12i)T \)
good7 \( 1 + (2.32 + 1.34i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-4.61 + 2.66i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.18 - 3.78i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.70 - 1.56i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.478 - 0.828i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.59 + 2.76i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.98iT - 31T^{2} \)
37 \( 1 + (4.20 - 2.42i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.78 + 2.18i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.620 + 1.07i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 2.61iT - 47T^{2} \)
53 \( 1 - 6.98T + 53T^{2} \)
59 \( 1 + (-5.03 - 2.90i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.42 - 2.47i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.34 - 0.778i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-8.89 - 5.13i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 0.569iT - 73T^{2} \)
79 \( 1 - 17.1T + 79T^{2} \)
83 \( 1 + 13.3iT - 83T^{2} \)
89 \( 1 + (0.243 - 0.140i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-10.4 - 6.01i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.879244889773573622526749330505, −8.581025590485838322875959716287, −7.46572394885184229410412104579, −6.64531380196855725131501692576, −6.32256787263226780342490159447, −5.50603901268422451611322906064, −4.02411761186798127487763331688, −3.36238653419728930171547152155, −1.72791385987071500488502959019, −0.791243778620755397797294788379, 0.837322630507555505173873403674, 2.26785443255854547174115553494, 3.36428926307057081846912389276, 4.08563922530813073627465641762, 5.18853244918337742001549188951, 6.26240835341330415670440137870, 6.76411754185779721428477573486, 7.71682182880125365459367892465, 8.891443720485389815264195339927, 9.312990111774903269121362186371

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