Properties

Label 2-1950-65.49-c1-0-31
Degree $2$
Conductor $1950$
Sign $-0.194 + 0.980i$
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 − 1.73i)7-s + 0.999·8-s + (0.499 − 0.866i)9-s + (5.59 − 3.23i)11-s + 0.999i·12-s + (3.46 − i)13-s − 1.99·14-s + (−0.5 − 0.866i)16-s + (3.46 + 2i)17-s − 0.999·18-s + (−6.46 − 3.73i)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 + (0.377 − 0.654i)7-s + 0.353·8-s + (0.166 − 0.288i)9-s + (1.68 − 0.974i)11-s + 0.288i·12-s + (0.960 − 0.277i)13-s − 0.534·14-s + (−0.125 − 0.216i)16-s + (0.840 + 0.485i)17-s − 0.235·18-s + (−1.48 − 0.856i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.194 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.194 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-0.194 + 0.980i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ -0.194 + 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.038060938\)
\(L(\frac12)\) \(\approx\) \(2.038060938\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.46 + i)T \)
good7 \( 1 + (-1 + 1.73i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-5.59 + 3.23i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3.46 - 2i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (6.46 + 3.73i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.23 - 1.86i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.133 - 0.232i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.73iT - 31T^{2} \)
37 \( 1 + (-4.59 - 7.96i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.73 + i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-10.3 - 5.96i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 3.53T + 47T^{2} \)
53 \( 1 + 0.928iT - 53T^{2} \)
59 \( 1 + (7.33 + 4.23i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.19 + 9i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.73 + 9.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (10.7 + 6.19i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 - 13.9T + 79T^{2} \)
83 \( 1 + 8.92T + 83T^{2} \)
89 \( 1 + (-0.464 + 0.267i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.267 - 0.464i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.965123227463815661877892575423, −8.277030519783435766738474704692, −7.73226691748985730761408868693, −6.52780036872982393880822504667, −6.08477507264267854478374263114, −4.47735190577552411028307541481, −3.81801583348656737454018452627, −3.07244799578369672266450081523, −1.65418911833453985795087634164, −0.919269934217965818059530015140, 1.39324590336104368282217687876, 2.33505684365482430501948004488, 4.03281687600474836194425551405, 4.21647730580915101101002468683, 5.65223315395042425828216020078, 6.22078502000088631904605058969, 7.13971222905672441254062968877, 7.932677129959987446307138287402, 8.802991314927614253550119100773, 9.090253395144012118971603135251

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