Properties

Label 2-1950-13.10-c1-0-1
Degree $2$
Conductor $1950$
Sign $-0.777 + 0.629i$
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 + (0.242 + 0.140i)7-s + 0.999i·8-s + (−0.499 + 0.866i)9-s + (0.663 − 0.383i)11-s + 0.999·12-s + (−2.77 + 2.30i)13-s − 0.280·14-s + (−0.5 − 0.866i)16-s + (−1.10 + 1.92i)17-s − 0.999i·18-s + (−4.57 − 2.63i)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 + (0.0917 + 0.0529i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (0.200 − 0.115i)11-s + 0.288·12-s + (−0.769 + 0.639i)13-s − 0.0749·14-s + (−0.125 − 0.216i)16-s + (−0.269 + 0.465i)17-s − 0.235i·18-s + (−1.04 − 0.605i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.777 + 0.629i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.777 + 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-0.777 + 0.629i$
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: \((2,\ 1950,\ (\ :1/2),\ -0.777 + 0.629i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1516531863\)
\(L(\frac12)\) \(\approx\) \(0.1516531863\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.77 - 2.30i)T \)
good7 \( 1 + (-0.242 - 0.140i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.663 + 0.383i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.10 - 1.92i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.57 + 2.63i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.725 + 1.25i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.03 + 5.24i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 3.28iT - 31T^{2} \)
37 \( 1 + (3.07 - 1.77i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.92 + 1.10i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.80 - 3.12i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 1.06iT - 47T^{2} \)
53 \( 1 - 3.28T + 53T^{2} \)
59 \( 1 + (4.32 + 2.49i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.773 - 1.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.43 - 3.71i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.09 + 1.21i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 14.2iT - 73T^{2} \)
79 \( 1 + 14.4T + 79T^{2} \)
83 \( 1 + 4.42iT - 83T^{2} \)
89 \( 1 + (4.60 - 2.65i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.09 - 0.633i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.619352450979025719542725154246, −8.765833967985624455987138536424, −8.366152774443347225142336895309, −7.33929458288012774180379814132, −6.64829198920316787324658853485, −5.78525393822921264553083869960, −4.74331190916128823422594001665, −4.06050773972134710350092765257, −2.72744006136172788069071055090, −1.77630710467572904469153441925, 0.06078804906728435947556577575, 1.54118430960403340960725005630, 2.46227542496620694498954978439, 3.42801271512830144911811162478, 4.48369826594362356740238711263, 5.60473694554956697947349865760, 6.55452290418272474137700395738, 7.39017493084279518533342238613, 7.895544703351618691118598787064, 8.786131378514742654952125974363

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