Properties

Label 2-1950-65.4-c1-0-15
Degree $2$
Conductor $1950$
Sign $-0.677 - 0.735i$
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 + (2.32 + 4.02i)7-s + 0.999·8-s + (0.499 + 0.866i)9-s + (3.81 + 2.20i)11-s − 0.999i·12-s + (−1.32 + 3.35i)13-s − 4.64·14-s + (−0.5 + 0.866i)16-s + (−3.46 + 2i)17-s − 0.999·18-s + (6.96 − 4.02i)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 + (0.877 + 1.52i)7-s + 0.353·8-s + (0.166 + 0.288i)9-s + (1.14 + 0.663i)11-s − 0.288i·12-s + (−0.366 + 0.930i)13-s − 1.24·14-s + (−0.125 + 0.216i)16-s + (−0.840 + 0.485i)17-s − 0.235·18-s + (1.59 − 0.922i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.677 - 0.735i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.677 - 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-0.677 - 0.735i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ -0.677 - 0.735i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.960092386\)
\(L(\frac12)\) \(\approx\) \(1.960092386\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.32 - 3.35i)T \)
good7 \( 1 + (-2.32 - 4.02i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-3.81 - 2.20i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (3.46 - 2i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-6.96 + 4.02i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.845 - 0.488i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.15 - 3.73i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.44iT - 31T^{2} \)
37 \( 1 + (1.89 - 3.28i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (6.31 + 3.64i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.620 + 0.358i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 9.75T + 47T^{2} \)
53 \( 1 + 13.5iT - 53T^{2} \)
59 \( 1 + (-1.88 + 1.09i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.73 + 6.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.912 + 1.58i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.88 - 3.97i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 4.36T + 73T^{2} \)
79 \( 1 - 14.9T + 79T^{2} \)
83 \( 1 + 3.51T + 83T^{2} \)
89 \( 1 + (7.07 + 4.08i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.91 - 11.9i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.167165598141037731550451483789, −8.918746702928164533026740498605, −8.033327350967850838644993533132, −7.14006830146499638248168439762, −6.48610450771706619912967612680, −5.33455072819961641140151118420, −4.82891607189675155307468820402, −3.81707990571778229703572103345, −2.38137917863710547163422267530, −1.63049433138051925642494887692, 0.828398917652786137410891022377, 1.52480289663271850324816430095, 2.96373516516642412627754848148, 3.75383324249149748895611571784, 4.50821273554505616838431891213, 5.62280257492731550090420616312, 6.92970550245167935241806747248, 7.47315579358811576027789165045, 8.111273555767693370280320491493, 8.925361683595156795880199570704

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