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Properties

Label 2-1950-65.4-c1-0-37
Degree $2$
Conductor $1950$
Sign $-0.838 + 0.545i$
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 + (−0.241 − 0.417i)7-s + 0.999·8-s + (0.499 + 0.866i)9-s + (−2.00 − 1.15i)11-s + 0.999i·12-s + (1.86 − 3.08i)13-s + 0.482·14-s + (−0.5 + 0.866i)16-s + (5.88 − 3.39i)17-s − 0.999·18-s + (−4.39 + 2.53i)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 + (−0.0911 − 0.157i)7-s + 0.353·8-s + (0.166 + 0.288i)9-s + (−0.605 − 0.349i)11-s + 0.288i·12-s + (0.516 − 0.856i)13-s + 0.128·14-s + (−0.125 + 0.216i)16-s + (1.42 − 0.824i)17-s − 0.235·18-s + (−1.00 + 0.582i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.838 + 0.545i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.838 + 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-0.838 + 0.545i$
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: \((2,\ 1950,\ (\ :1/2),\ -0.838 + 0.545i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2604311287\)
\(L(\frac12)\) \(\approx\) \(0.2604311287\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.86 + 3.08i)T \)
good7 \( 1 + (0.241 + 0.417i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.00 + 1.15i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-5.88 + 3.39i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.39 - 2.53i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.98 + 3.45i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.20 - 7.28i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.952iT - 31T^{2} \)
37 \( 1 + (3.09 - 5.35i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-7.55 - 4.36i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-9.78 + 5.64i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 5.77T + 47T^{2} \)
53 \( 1 - 5.81iT - 53T^{2} \)
59 \( 1 + (10.2 - 5.91i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.68 + 8.11i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.69 + 6.40i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.01 + 1.74i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 8.53T + 73T^{2} \)
79 \( 1 + 9.03T + 79T^{2} \)
83 \( 1 + 7.94T + 83T^{2} \)
89 \( 1 + (7.80 + 4.50i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.59 + 7.95i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.672281459365761010611353639604, −7.901316265722269251154697267091, −7.48421312990826742985231366552, −6.39255628927785016904188819563, −5.78427269857748802150034447690, −5.15437207208304197491377056084, −4.00146049389476964715317435383, −2.85649368803266546950940416627, −1.37364431396466660401782645187, −0.11983315231397819867503609798, 1.50595368305783193592166956480, 2.53801010624945340181366956228, 3.85893626919207135115475463156, 4.31996779606991959977361289698, 5.63037198923280564161741794171, 6.11811811783582805427127899928, 7.35262722736875999281314036307, 7.989604055401268914587447207742, 8.913697954342627848412447391975, 9.659818643451747445194006163877

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