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Properties

Label 2-1950-13.4-c1-0-37
Degree $2$
Conductor $1950$
Sign $0.751 + 0.659i$
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 + (1.32 − 0.763i)7-s + 0.999i·8-s + (−0.499 − 0.866i)9-s + (−1.14 − 0.658i)11-s + 0.999·12-s + (−2.67 − 2.41i)13-s + 1.52·14-s + (−0.5 + 0.866i)16-s + (0.784 + 1.35i)17-s − 0.999i·18-s + (4.18 − 2.41i)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 + (0.500 − 0.288i)7-s + 0.353i·8-s + (−0.166 − 0.288i)9-s + (−0.343 − 0.198i)11-s + 0.288·12-s + (−0.743 − 0.669i)13-s + 0.408·14-s + (−0.125 + 0.216i)16-s + (0.190 + 0.329i)17-s − 0.235i·18-s + (0.959 − 0.553i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 + 0.659i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.751 + 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.751 + 0.659i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ 0.751 + 0.659i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.786703316\)
\(L(\frac12)\) \(\approx\) \(2.786703316\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.67 + 2.41i)T \)
good7 \( 1 + (-1.32 + 0.763i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.14 + 0.658i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.784 - 1.35i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.18 + 2.41i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.22 + 7.31i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.21 + 3.83i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.62iT - 31T^{2} \)
37 \( 1 + (-2.42 - 1.40i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.35 - 0.784i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.64 + 4.58i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 4.94iT - 47T^{2} \)
53 \( 1 - 13.9T + 53T^{2} \)
59 \( 1 + (-9.07 + 5.23i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.49 - 4.31i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.40 + 1.38i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (12.8 - 7.41i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 5.98iT - 73T^{2} \)
79 \( 1 + 4.87T + 79T^{2} \)
83 \( 1 + 6.39iT - 83T^{2} \)
89 \( 1 + (15.9 + 9.22i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.66 + 0.963i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.737322902509523985386735417882, −8.234399173605950964144441087896, −7.35843795244133462000109970541, −6.92022945709390264049954075136, −5.82789772653165006927754710455, −5.09396789527210311120764300732, −4.28488100410440907386401587545, −3.08570301965752862058455264181, −2.39441127687976833115796483042, −0.835861275502539360584559044417, 1.44308348632847872523988204408, 2.56368128956498045575356012160, 3.39828844050068084424760438815, 4.40061154639300818902118750016, 5.16381463349132059502072127354, 5.69058953591766395011972152257, 7.03944531956675329144448795771, 7.59028802963538550585325590438, 8.619597668020593026514115747251, 9.506480238639276884776554078355

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