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Properties

Label 2-1950-65.9-c1-0-1
Degree $2$
Conductor $1950$
Sign $-0.820 - 0.572i$
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 + (1.73 − i)7-s − 0.999i·8-s + (0.499 + 0.866i)9-s + (−1 + 1.73i)11-s + 0.999i·12-s + (−2.59 + 2.5i)13-s − 1.99·14-s + (−0.5 + 0.866i)16-s + (−4.33 + 2.5i)17-s − 0.999i·18-s + (−1 − 1.73i)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 + (0.654 − 0.377i)7-s − 0.353i·8-s + (0.166 + 0.288i)9-s + (−0.301 + 0.522i)11-s + 0.288i·12-s + (−0.720 + 0.693i)13-s − 0.534·14-s + (−0.125 + 0.216i)16-s + (−1.05 + 0.606i)17-s − 0.235i·18-s + (−0.229 − 0.397i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.820 - 0.572i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.820 - 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-0.820 - 0.572i$
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: \((2,\ 1950,\ (\ :1/2),\ -0.820 - 0.572i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4615745325\)
\(L(\frac12)\) \(\approx\) \(0.4615745325\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.59 - 2.5i)T \)
good7 \( 1 + (-1.73 + i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (4.33 - 2.5i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.19 + 3i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.5 - 7.79i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (9.52 + 5.5i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.5 - 4.33i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-8.66 + 5i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 2iT - 47T^{2} \)
53 \( 1 + iT - 53T^{2} \)
59 \( 1 + (4 + 6.92i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.5 - 9.52i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.73 - i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-7 - 12.1i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 13iT - 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 6iT - 83T^{2} \)
89 \( 1 + (-1 + 1.73i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.73 + i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.327299271955976713581550494189, −8.878152185997758900313546834796, −8.071214113707389444054169378377, −7.29096736512624528617977635040, −6.71252635208430678622137515276, −5.30218831435858337224796567558, −4.41235510107033913759963203982, −3.73037395961379360625968847852, −2.33228084191084911806414021304, −1.78066729123762837404490369090, 0.17225415088682102828990645716, 1.80728138493685538605414544404, 2.56215244996036002473197197903, 3.81435204865125962142531271564, 5.01582029689882105670062074960, 5.73946222763695844080875234188, 6.64702588859483034623984913483, 7.65706384904069473983801997296, 7.968186421018230414667153179630, 8.781419066586178115389846856731

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