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Properties

Label 2-1890-63.25-c1-0-7
Degree $2$
Conductor $1890$
Sign $0.243 - 0.969i$
Analytic cond. $15.0917$
Root an. cond. $3.88480$
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  − 2-s + 4-s + (0.5 − 0.866i)5-s + (1.85 + 1.88i)7-s − 8-s + (−0.5 + 0.866i)10-s + (1.39 + 2.41i)11-s + (0.900 + 1.55i)13-s + (−1.85 − 1.88i)14-s + 16-s + (0.292 − 0.506i)17-s + (2.54 + 4.40i)19-s + (0.5 − 0.866i)20-s + (−1.39 − 2.41i)22-s + (−1.60 + 2.78i)23-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + (0.223 − 0.387i)5-s + (0.699 + 0.714i)7-s − 0.353·8-s + (−0.158 + 0.273i)10-s + (0.420 + 0.727i)11-s + (0.249 + 0.432i)13-s + (−0.494 − 0.505i)14-s + 0.250·16-s + (0.0708 − 0.122i)17-s + (0.583 + 1.01i)19-s + (0.111 − 0.193i)20-s + (−0.297 − 0.514i)22-s + (−0.335 + 0.580i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.243 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.243 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1890\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $0.243 - 0.969i$
Analytic conductor: \(15.0917\)
Root analytic conductor: \(3.88480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1890} (991, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1890,\ (\ :1/2),\ 0.243 - 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.296615364\)
\(L(\frac12)\) \(\approx\) \(1.296615364\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-1.85 - 1.88i)T \)
good11 \( 1 + (-1.39 - 2.41i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.900 - 1.55i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.292 + 0.506i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.54 - 4.40i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.60 - 2.78i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.226 - 0.392i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 7.90T + 31T^{2} \)
37 \( 1 + (-1.10 - 1.91i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.43 - 4.21i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.41 + 4.17i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 10.7T + 47T^{2} \)
53 \( 1 + (2.04 - 3.54i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 7.99T + 59T^{2} \)
61 \( 1 - 15.1T + 61T^{2} \)
67 \( 1 + 13.9T + 67T^{2} \)
71 \( 1 - 5.78T + 71T^{2} \)
73 \( 1 + (-3.03 + 5.26i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 0.652T + 79T^{2} \)
83 \( 1 + (1.43 - 2.48i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.58 + 2.74i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.00 + 8.66i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.405732685044773790110762383699, −8.673193792324493803029075237332, −7.913967964041915477253048026538, −7.25242104973045107750402746107, −6.18854656673927202142271523295, −5.48350332735543149269710147365, −4.56291465898704100727966568621, −3.43516120546702991688025038782, −2.04263601738195042116707798670, −1.42435980800225118959061543729, 0.62454166663152305202506656083, 1.78592795205432323109280673892, 3.00966652461654552599777125625, 3.93349582811989259684132171471, 5.09930561719631491991603723908, 6.01010278387379030561274628467, 6.86049944857416010531488631512, 7.55526887986255959616193690444, 8.282793370991290216962597643945, 9.032895321817797690440076206330

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