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Properties

Label 2-1950-13.4-c1-0-15
Degree $2$
Conductor $1950$
Sign $0.890 + 0.454i$
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 + (−4.44 + 2.56i)7-s − 0.999i·8-s + (−0.499 − 0.866i)9-s + (−4.83 − 2.78i)11-s + 0.999·12-s + (3.19 + 1.67i)13-s + 5.13·14-s + (−0.5 + 0.866i)16-s + (−0.641 − 1.11i)17-s + 0.999i·18-s + (5.75 − 3.32i)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 + (−1.67 + 0.969i)7-s − 0.353i·8-s + (−0.166 − 0.288i)9-s + (−1.45 − 0.841i)11-s + 0.288·12-s + (0.884 + 0.465i)13-s + 1.37·14-s + (−0.125 + 0.216i)16-s + (−0.155 − 0.269i)17-s + 0.235i·18-s + (1.32 − 0.762i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 + 0.454i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.890 + 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.890 + 0.454i$
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: \((2,\ 1950,\ (\ :1/2),\ 0.890 + 0.454i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9089497960\)
\(L(\frac12)\) \(\approx\) \(0.9089497960\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.19 - 1.67i)T \)
good7 \( 1 + (4.44 - 2.56i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (4.83 + 2.78i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.641 + 1.11i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.75 + 3.32i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.43 - 5.94i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.75 - 3.03i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.30iT - 31T^{2} \)
37 \( 1 + (-7.25 - 4.19i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.11 - 0.641i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.17 - 3.77i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 5.02iT - 47T^{2} \)
53 \( 1 - 6.30T + 53T^{2} \)
59 \( 1 + (-1.31 + 0.759i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.19 - 8.99i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.48 - 1.43i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-9.44 + 5.45i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 4.94iT - 73T^{2} \)
79 \( 1 - 10.1T + 79T^{2} \)
83 \( 1 + 17.3iT - 83T^{2} \)
89 \( 1 + (-5.57 - 3.22i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.20 - 3.58i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.348162960551470833428799171108, −8.418161811036620022814267845399, −7.71861542011567511717448552325, −6.86180323483510670008385785469, −5.99568038023525309800251858936, −5.44167566077377152851598849845, −3.66515010877363850751853844586, −2.98862037642182803206450703838, −2.32196292710966639777897488221, −0.68911200582651171364615620842, 0.63636203210231227061436846520, 2.42049515565253284001058717361, 3.37498063540502031907512028779, 4.19304444269923943747199331280, 5.40098661863102106156301014996, 6.15989739687322336158247743918, 7.03082502749057274474924944158, 7.74856676705870695437264305619, 8.371261638162137119639644576828, 9.542738629391812653690263885565

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