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Properties

Label 2-1950-65.9-c1-0-16
Degree $2$
Conductor $1950$
Sign $0.976 - 0.216i$
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 + (1.00 − 0.581i)7-s − 0.999i·8-s + (0.499 + 0.866i)9-s + (−0.5 + 0.866i)11-s − 0.999i·12-s + (3.60 − 0.0811i)13-s − 1.16·14-s + (−0.5 + 0.866i)16-s + (1.00 − 0.581i)17-s − 0.999i·18-s + (2.58 + 4.47i)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 + (0.380 − 0.219i)7-s − 0.353i·8-s + (0.166 + 0.288i)9-s + (−0.150 + 0.261i)11-s − 0.288i·12-s + (0.999 − 0.0225i)13-s − 0.310·14-s + (−0.125 + 0.216i)16-s + (0.244 − 0.140i)17-s − 0.235i·18-s + (0.592 + 1.02i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.216i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.976 - 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.976 - 0.216i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (1699, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ 0.976 - 0.216i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.066762831\)
\(L(\frac12)\) \(\approx\) \(1.066762831\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.60 + 0.0811i)T \)
good7 \( 1 + (-1.00 + 0.581i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.00 + 0.581i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.58 - 4.47i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.87 + 1.08i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.16 - 7.20i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.837T + 31T^{2} \)
37 \( 1 + (0.140 + 0.0811i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.581 + 1.00i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.73 + 1.58i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 6iT - 47T^{2} \)
53 \( 1 - 7.48iT - 53T^{2} \)
59 \( 1 + (-5 - 8.66i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.0811 - 0.140i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.24 + 9.08i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 3.32iT - 73T^{2} \)
79 \( 1 - 13.4T + 79T^{2} \)
83 \( 1 - 9iT - 83T^{2} \)
89 \( 1 + (5.74 - 9.94i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.61 - 2.66i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.219990950684407718957411582342, −8.420883399515906191874440087460, −7.67738588896990101654446744833, −7.06137359064017692084040770872, −6.04830792540880185066801368821, −5.32958472020344650507406844073, −4.16525333628754699448802349592, −3.27972287498416944747477379105, −1.91404884859560933572301755164, −1.04205468015676740365987825196, 0.63018857481576998454019678405, 1.92499625582596355159986409705, 3.28903498507064449545022946998, 4.37521370121521851338907870378, 5.36309429613189061540723435823, 5.96818362029827893521111803789, 6.76906084520999745679550874056, 7.72245494144799836639898237702, 8.352853979894070249646718387335, 9.197679246167833518322126869299

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