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Properties

Label 2-1950-65.4-c1-0-29
Degree $2$
Conductor $1950$
Sign $-0.193 + 0.981i$
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 + (1.82 + 3.15i)7-s − 0.999·8-s + (0.499 + 0.866i)9-s + (−1.44 − 0.834i)11-s + 0.999i·12-s + (−2.82 − 2.24i)13-s + 3.64·14-s + (−0.5 + 0.866i)16-s + (3.46 − 2i)17-s + 0.999·18-s + (−5.46 + 3.15i)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 + (0.688 + 1.19i)7-s − 0.353·8-s + (0.166 + 0.288i)9-s + (−0.435 − 0.251i)11-s + 0.288i·12-s + (−0.782 − 0.622i)13-s + 0.974·14-s + (−0.125 + 0.216i)16-s + (0.840 − 0.485i)17-s + 0.235·18-s + (−1.25 + 0.724i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.193 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.193 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-0.193 + 0.981i$
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: \((2,\ 1950,\ (\ :1/2),\ -0.193 + 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.556799442\)
\(L(\frac12)\) \(\approx\) \(1.556799442\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.82 + 2.24i)T \)
good7 \( 1 + (-1.82 - 3.15i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.44 + 0.834i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-3.46 + 2i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.46 - 3.15i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.07 - 0.622i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.02 + 8.69i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 4.21iT - 31T^{2} \)
37 \( 1 + (-4.93 + 8.54i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-8.04 - 4.64i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6.55 + 3.78i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 6.82T + 47T^{2} \)
53 \( 1 + 0.848iT - 53T^{2} \)
59 \( 1 + (5.29 - 3.05i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.73 + 6.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.37 + 12.7i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.04 - 1.75i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 12.2T + 73T^{2} \)
79 \( 1 + 9.93T + 79T^{2} \)
83 \( 1 - 7.95T + 83T^{2} \)
89 \( 1 + (5.15 + 2.97i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.37 - 2.38i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.102304471107947249044340146979, −8.024349430623565442623419456796, −7.66913099513027353547266086863, −6.14987902116854408475846430630, −5.74257128193835088254615466505, −4.97458163480038923090633294009, −4.10909304993527156676362150882, −2.65726511923598963794165809039, −2.17615795323868133415574850196, −0.62208951802235818498261402872, 1.11416443403744275024905446731, 2.70735156744703368482756098974, 4.04148011256736093464000046673, 4.59296292788968171039443973151, 5.21065102655243761203410973679, 6.32245885841686172305871961579, 7.04111069403980366278933688980, 7.61550086177992469066879429205, 8.489312134967290379974074812968, 9.346531102500396825260527864581

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