Properties

Label 2-1950-65.4-c1-0-31
Degree $2$
Conductor $1950$
Sign $0.194 + 0.980i$
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 − 1.73i)7-s − 0.999·8-s + (0.499 + 0.866i)9-s + (0.401 + 0.232i)11-s − 0.999i·12-s + (3.46 + i)13-s − 1.99·14-s + (−0.5 + 0.866i)16-s + (3.46 − 2i)17-s + 0.999·18-s + (0.464 − 0.267i)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 + (−0.377 − 0.654i)7-s − 0.353·8-s + (0.166 + 0.288i)9-s + (0.121 + 0.0699i)11-s − 0.288i·12-s + (0.960 + 0.277i)13-s − 0.534·14-s + (−0.125 + 0.216i)16-s + (0.840 − 0.485i)17-s + 0.235·18-s + (0.106 − 0.0614i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.194 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.194 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.194 + 0.980i$
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: \((2,\ 1950,\ (\ :1/2),\ 0.194 + 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.447862364\)
\(L(\frac12)\) \(\approx\) \(2.447862364\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.46 - i)T \)
good7 \( 1 + (1 + 1.73i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.401 - 0.232i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-3.46 + 2i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.464 + 0.267i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.232 + 0.133i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.86 + 3.23i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.73iT - 31T^{2} \)
37 \( 1 + (-0.598 + 1.03i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.73 + i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.66 - 0.964i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 10.4T + 47T^{2} \)
53 \( 1 + 12.9iT - 53T^{2} \)
59 \( 1 + (-1.33 + 0.767i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.19 + 9i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.26 + 3.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (7.26 - 4.19i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 - 0.0717T + 79T^{2} \)
83 \( 1 + 4.92T + 83T^{2} \)
89 \( 1 + (6.46 + 3.73i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.73 - 6.46i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.172333882051628320948580547842, −8.378104467242612938385397414113, −7.49649722130559104963996387809, −6.60549297472077128418038741697, −5.70392218017096578726782762106, −4.70788250850072145526344219263, −3.82719395429385342031519186848, −3.28814083478708514166512010591, −2.12257092624355351302241163566, −0.843730638254039619293080474540, 1.29614736874275269522506495524, 2.75919996532013318263877414477, 3.47568547203510542941806165865, 4.42734392707080014134904902086, 5.69525538287290360196218755208, 6.01717779587538313902027522957, 7.03635715267163895107408521299, 7.74625676266327011517617449352, 8.638474524931859060831347745844, 8.952637087748687763246498251545

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