↑

Properties

Label 2-1950-13.3-c1-0-14
Degree $2$
Conductor $1950$
Sign $-0.981 - 0.189i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + (−0.499 + 0.866i)6-s + (−0.366 + 0.633i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (1.36 + 2.36i)11-s − 0.999·12-s + (3.23 + 1.59i)13-s − 0.732·14-s + (−0.5 − 0.866i)16-s + (−1.23 + 2.13i)17-s − 0.999·18-s + (−2.36 + 4.09i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.204 + 0.353i)6-s + (−0.138 + 0.239i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (0.411 + 0.713i)11-s − 0.288·12-s + (0.896 + 0.443i)13-s − 0.195·14-s + (−0.125 − 0.216i)16-s + (−0.298 + 0.517i)17-s − 0.235·18-s + (−0.542 + 0.940i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.189i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.981 - 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-0.981 - 0.189i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ -0.981 - 0.189i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.868036324\)
\(L(\frac12)\) \(\approx\) \(1.868036324\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 \)
13 \( 1 + (-3.23 - 1.59i)T \)
good7 \( 1 + (0.366 - 0.633i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.36 - 2.36i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.23 - 2.13i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.36 - 4.09i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.09 + 3.63i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.232 + 0.401i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + (2.96 + 5.13i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.598 + 1.03i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.36 - 5.83i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 9.66T + 47T^{2} \)
53 \( 1 + 4.26T + 53T^{2} \)
59 \( 1 + (4.19 - 7.26i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.06 - 12.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.83 + 8.36i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.36 - 4.09i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 12.6T + 73T^{2} \)
79 \( 1 - 12T + 79T^{2} \)
83 \( 1 + 8.73T + 83T^{2} \)
89 \( 1 + (4.46 + 7.73i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5 - 8.66i)T + (-48.5 - 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.312231306889710573603983730239, −8.744477618307031779917490878162, −8.067077753144484784360256527194, −7.14621536826823701297350090324, −6.22979378631966223500489896304, −5.75335477443050846171578513844, −4.32625133698592226968269237333, −4.20153470649733930268900162412, −2.96821224236244426968215427557, −1.74336562978274807672317519686, 0.57354921231064176360029293585, 1.72032277675907784967917181176, 2.91846206950680049868949572002, 3.59286117092053039049998725854, 4.58128951033937909682282506261, 5.64475712212537717246223483151, 6.40021508065832168982600818412, 7.14621140086776856155604142176, 8.242221128265142177597948291547, 8.795666205236255797659058300191

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