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Properties

Label 2-1950-65.4-c1-0-5
Degree $2$
Conductor $1950$
Sign $0.668 - 0.743i$
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 + (5.59 + 3.23i)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 + (−6.46 + 3.73i)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 + (1.68 + 0.974i)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 + (−1.48 + 0.856i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.668 - 0.743i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.668 - 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.668 - 0.743i$
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: \((2,\ 1950,\ (\ :1/2),\ 0.668 - 0.743i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8387491663\)
\(L(\frac12)\) \(\approx\) \(0.8387491663\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.59 - 3.23i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (3.46 - 2i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (6.46 - 3.73i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.23 - 1.86i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.133 + 0.232i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.73iT - 31T^{2} \)
37 \( 1 + (4.59 - 7.96i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.73 - i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (10.3 - 5.96i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 3.53T + 47T^{2} \)
53 \( 1 + 0.928iT - 53T^{2} \)
59 \( 1 + (7.33 - 4.23i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.19 - 9i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.73 + 9.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (10.7 - 6.19i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 - 13.9T + 79T^{2} \)
83 \( 1 - 8.92T + 83T^{2} \)
89 \( 1 + (-0.464 - 0.267i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.267 - 0.464i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.505154858556167190805414917648, −8.666606690362961675901730746998, −7.56346733718120231409260640116, −6.57953954075075180247355715755, −6.41991099937697517677493964814, −5.02004666964420272589637470491, −4.31951825423691141629881022692, −3.62186847667044757490827496112, −2.19981619376200398616857544961, −1.30917552747722356706707280976, 0.29446700082279174144960222385, 2.22130277008658820312603057110, 3.41533533101684018299629715098, 4.33377224127870521611019140209, 5.04663053951042748518829427926, 6.01400106416086191009702261847, 6.66275620655224731353758926545, 7.07648183016529583470832214536, 8.595757703777057604668975202001, 8.946709088904441887645488889963

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