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Properties

Label 2-1950-65.29-c1-0-30
Degree $2$
Conductor $1950$
Sign $0.343 + 0.939i$
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 + (2.73 + 1.58i)7-s − 0.999i·8-s + (0.499 − 0.866i)9-s + (−1.5 − 2.59i)11-s − 0.999i·12-s + (1.87 + 3.08i)13-s + 3.16·14-s + (−0.5 − 0.866i)16-s + (−6.20 − 3.58i)17-s − 0.999i·18-s + (1.58 − 2.73i)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 + (1.03 + 0.597i)7-s − 0.353i·8-s + (0.166 − 0.288i)9-s + (−0.452 − 0.783i)11-s − 0.288i·12-s + (0.519 + 0.854i)13-s + 0.845·14-s + (−0.125 − 0.216i)16-s + (−1.50 − 0.868i)17-s − 0.235i·18-s + (0.362 − 0.628i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.343 + 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.343 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.343 + 0.939i$
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: \((2,\ 1950,\ (\ :1/2),\ 0.343 + 0.939i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.335776548\)
\(L(\frac12)\) \(\approx\) \(3.335776548\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.73 - 1.58i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (6.20 + 3.58i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.58 + 2.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.60 + 2.08i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.16 + 7.20i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 9.16T + 31T^{2} \)
37 \( 1 + (-9.08 + 5.24i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.74 - 8.21i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.47 - 2.58i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 6iT - 47T^{2} \)
53 \( 1 + 7.16iT - 53T^{2} \)
59 \( 1 + (3 - 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.24 - 12.5i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.46 + 2i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.91 + 6.78i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 1.32iT - 73T^{2} \)
79 \( 1 + 9.16T + 79T^{2} \)
83 \( 1 - 13.6iT - 83T^{2} \)
89 \( 1 + (3.58 + 6.20i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.04 + 2.33i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.039995361728850374951689296778, −8.301424316516219454490199239863, −7.52758371301714011175316883205, −6.51353750141538858874449355160, −5.84598586834302702238746879828, −4.69343328526035967403862435369, −4.28622678837671406824678482222, −2.76621619774219096351730718806, −2.37927003529498807738094578383, −1.01268551122125552924945014874, 1.48389659182803053037959182007, 2.60729152937320153518753359152, 3.71138622993894337879499529405, 4.47899558432838520448646189438, 5.10088107918929474754877995239, 6.08993874185819157882340382362, 7.11211097312678731744367648880, 7.78288801235820890454389509225, 8.341472761824728216662247323342, 9.165649692939228526320892296298

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