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Properties

Label 2-1950-65.49-c1-0-38
Degree $2$
Conductor $1950$
Sign $-0.997 + 0.0663i$
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 + (0.465 − 0.807i)7-s + 0.999·8-s + (0.499 − 0.866i)9-s + (1.45 − 0.841i)11-s + 0.999i·12-s + (−3.08 + 1.86i)13-s − 0.931·14-s + (−0.5 − 0.866i)16-s + (−2.21 − 1.27i)17-s − 0.999·18-s + (−2.27 − 1.31i)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 + (0.176 − 0.305i)7-s + 0.353·8-s + (0.166 − 0.288i)9-s + (0.439 − 0.253i)11-s + 0.288i·12-s + (−0.856 + 0.516i)13-s − 0.249·14-s + (−0.125 − 0.216i)16-s + (−0.536 − 0.309i)17-s − 0.235·18-s + (−0.522 − 0.301i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0663i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0663i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-0.997 + 0.0663i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ -0.997 + 0.0663i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8798618846\)
\(L(\frac12)\) \(\approx\) \(0.8798618846\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.08 - 1.86i)T \)
good7 \( 1 + (-0.465 + 0.807i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.45 + 0.841i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.21 + 1.27i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.27 + 1.31i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.59 - 3.22i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.69 + 2.93i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 9.29iT - 31T^{2} \)
37 \( 1 + (4.57 + 7.93i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.01 + 0.587i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.55 + 1.47i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 3.97T + 47T^{2} \)
53 \( 1 + 5.36iT - 53T^{2} \)
59 \( 1 + (-3.44 - 1.98i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.36 + 7.55i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.59 + 4.49i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.40 - 1.96i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 7.26T + 73T^{2} \)
79 \( 1 - 3.68T + 79T^{2} \)
83 \( 1 + 3.84T + 83T^{2} \)
89 \( 1 + (11.6 - 6.73i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.04 + 12.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.983282468605142729573986810286, −8.050523047407857281865075982167, −7.43691346928987604569563773237, −6.65524022665495750200938330315, −5.58374797222167004478380178332, −4.31276478525305847841452760657, −3.82300592060800695784946212418, −2.49530030991941819299684207901, −1.85309652170853325290252707327, −0.31712907150854309273136301090, 1.61911903016887804458634055592, 2.69983677143455924130045366091, 3.93073215997405852257587657839, 4.76877734601755852802846153297, 5.57735314998747876911221415214, 6.59643528324174233549992285128, 7.21834359981582402325583319936, 8.261880669706747739240354521081, 8.578471136928574528091864415081, 9.430826116948136935457923341207

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