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Properties

Label 2-1950-65.29-c1-0-24
Degree $2$
Conductor $1950$
Sign $0.957 + 0.288i$
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 + (0.866 − 0.5i)3-s + (0.499 − 0.866i)4-s + (0.499 − 0.866i)6-s + (4.47 + 2.58i)7-s − 0.999i·8-s + (0.499 − 0.866i)9-s + (−0.5 − 0.866i)11-s − 0.999i·12-s + (1.87 + 3.08i)13-s + 5.16·14-s + (−0.5 − 0.866i)16-s + (4.47 + 2.58i)17-s − 0.999i·18-s + (−0.581 + 1.00i)19-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.499 − 0.288i)3-s + (0.249 − 0.433i)4-s + (0.204 − 0.353i)6-s + (1.68 + 0.975i)7-s − 0.353i·8-s + (0.166 − 0.288i)9-s + (−0.150 − 0.261i)11-s − 0.288i·12-s + (0.519 + 0.854i)13-s + 1.37·14-s + (−0.125 − 0.216i)16-s + (1.08 + 0.626i)17-s − 0.235i·18-s + (−0.133 + 0.230i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.288i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 + 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.957 + 0.288i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (1849, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ 0.957 + 0.288i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.735836168\)
\(L(\frac12)\) \(\approx\) \(3.735836168\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 \)
13 \( 1 + (-1.87 - 3.08i)T \)
good7 \( 1 + (-4.47 - 2.58i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-4.47 - 2.58i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.581 - 1.00i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.60 - 2.08i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.16 - 3.74i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 7.16T + 31T^{2} \)
37 \( 1 + (5.33 - 3.08i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.58 + 4.47i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.73 - 1.58i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 6iT - 47T^{2} \)
53 \( 1 + 11.4iT - 53T^{2} \)
59 \( 1 + (-5 + 8.66i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.08 - 5.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.24 + 7.34i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 9.32iT - 73T^{2} \)
79 \( 1 + 5.48T + 79T^{2} \)
83 \( 1 - 9iT - 83T^{2} \)
89 \( 1 + (-3.74 - 6.48i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.34 + 3.66i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.937326825706588730118711324430, −8.398496889056116080842532338567, −7.75005671669057468269166389624, −6.72974262225744244932363446016, −5.66070840354109154687518153259, −5.21026461791967083270046926959, −4.11168220590802833849034717622, −3.29457328807210026436707551534, −1.95123151662546194562566207789, −1.60124901110636547712790806918, 1.20592681270381483151500825509, 2.44201183037425136157062243092, 3.61466158501532225536387648791, 4.32717573693437045227334411759, 5.08397165813977897509252861226, 5.79207192011194611199822738217, 7.10719559810405457427001530664, 7.74315733170156254721619885214, 8.117961295479890953511618788199, 9.023637179733030467047912463920

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