Properties

Label 2-1950-65.29-c1-0-38
Degree $2$
Conductor $1950$
Sign $-0.997 - 0.0727i$
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 + (−4.33 − 2.5i)7-s − 0.999i·8-s + (0.499 − 0.866i)9-s + (1.5 + 2.59i)11-s − 0.999i·12-s + (2.59 − 2.5i)13-s − 5·14-s + (−0.5 − 0.866i)16-s + (−6.92 − 4i)17-s − 0.999i·18-s + (−2.5 + 4.33i)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 + (−1.63 − 0.944i)7-s − 0.353i·8-s + (0.166 − 0.288i)9-s + (0.452 + 0.783i)11-s − 0.288i·12-s + (0.720 − 0.693i)13-s − 1.33·14-s + (−0.125 − 0.216i)16-s + (−1.68 − 0.970i)17-s − 0.235i·18-s + (−0.573 + 0.993i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0727i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-0.997 - 0.0727i$
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: \((2,\ 1950,\ (\ :1/2),\ -0.997 - 0.0727i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.291116870\)
\(L(\frac12)\) \(\approx\) \(1.291116870\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.33 + 2.5i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (6.92 + 4i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.5 - 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.46 - 2i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2 + 3.46i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 + (-6.06 + 3.5i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (3 + 5.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.19 + 3i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 3iT - 47T^{2} \)
53 \( 1 - iT - 53T^{2} \)
59 \( 1 + (-6 + 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.92 - 4i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (1 - 1.73i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 2T + 79T^{2} \)
83 \( 1 - 8iT - 83T^{2} \)
89 \( 1 + (5.5 + 9.52i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (48.5 + 84.0i)T^{2} \)
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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

−9.001911329421412014788667432173, −7.86169829180242070815817508636, −6.97814861597013518270604257357, −6.52586152397616756576179201301, −5.69652447229857975633277681702, −4.21946504993911972273580217295, −3.86032523594997992433360251479, −2.90122414369914056331257572451, −1.86096082105633008212170280385, −0.31991586017415356849037143315, 2.09261482577882818190885502860, 3.00569491303317172080502338235, 3.78782693813921073608977186146, 4.54031148420467874918345474758, 5.83183531361651589813620540873, 6.46146716023723279593934913335, 6.76596112803999286711129052905, 8.275110337199283842881027786942, 8.880312835240458803524501507339, 9.207021738278270607587828971558

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