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Properties

Label 2-1950-13.4-c1-0-32
Degree $2$
Conductor $1950$
Sign $-0.947 + 0.319i$
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 + (0.5 − 0.866i)3-s + (0.499 + 0.866i)4-s + (−0.866 + 0.499i)6-s + (−2.42 + 1.40i)7-s − 0.999i·8-s + (−0.499 − 0.866i)9-s + (−0.515 − 0.297i)11-s + 0.999·12-s + (−3.43 + 1.10i)13-s + 2.80·14-s + (−0.5 + 0.866i)16-s + (2.87 + 4.98i)17-s + 0.999i·18-s + (6.59 − 3.80i)19-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.288 − 0.499i)3-s + (0.249 + 0.433i)4-s + (−0.353 + 0.204i)6-s + (−0.918 + 0.530i)7-s − 0.353i·8-s + (−0.166 − 0.288i)9-s + (−0.155 − 0.0896i)11-s + 0.288·12-s + (−0.951 + 0.307i)13-s + 0.749·14-s + (−0.125 + 0.216i)16-s + (0.697 + 1.20i)17-s + 0.235i·18-s + (1.51 − 0.873i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.947 + 0.319i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.947 + 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-0.947 + 0.319i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ -0.947 + 0.319i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5447273832\)
\(L(\frac12)\) \(\approx\) \(0.5447273832\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 \)
13 \( 1 + (3.43 - 1.10i)T \)
good7 \( 1 + (2.42 - 1.40i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.515 + 0.297i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.87 - 4.98i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.59 + 3.80i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.32 + 4.02i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.26 + 2.18i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.59iT - 31T^{2} \)
37 \( 1 + (8.98 + 5.18i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.98 + 2.87i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.12 - 3.67i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 2.89iT - 47T^{2} \)
53 \( 1 + 13.8T + 53T^{2} \)
59 \( 1 + (8.40 - 4.85i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.41 + 5.91i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.80 + 3.93i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.11 - 0.642i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 14.5iT - 73T^{2} \)
79 \( 1 - 1.83T + 79T^{2} \)
83 \( 1 + 4.19iT - 83T^{2} \)
89 \( 1 + (-5.24 - 3.02i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-14.6 + 8.45i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.122409178552059151122095959119, −8.002173112596673202188190054166, −7.46867165342636215632889719455, −6.58892197165017725799620861311, −5.86272102933576407769275477337, −4.72732740946980354993192235015, −3.37135491440576300826473013850, −2.79023674880984733008647654643, −1.71517770897058610745244537836, −0.23923948624478640300987742779, 1.31632007630023872666498444117, 3.00916407726727679808673192189, 3.40332965128446644965333016218, 4.99033425948421854731906845932, 5.35584243452377777811925018337, 6.64587099085359879524499913703, 7.32884222110413688112863437439, 7.84201495348828173950774873386, 8.926339688950830315707775480412, 9.646301569574348544809284170734

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