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Properties

Label 2-1890-63.38-c1-0-24
Degree $2$
Conductor $1890$
Sign $0.578 + 0.815i$
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  + i·2-s − 4-s + (0.5 − 0.866i)5-s + (0.408 − 2.61i)7-s − i·8-s + (0.866 + 0.5i)10-s + (−3.42 + 1.97i)11-s + (2.82 − 1.63i)13-s + (2.61 + 0.408i)14-s + 16-s + (0.497 − 0.861i)17-s + (−4.90 + 2.83i)19-s + (−0.5 + 0.866i)20-s + (−1.97 − 3.42i)22-s + (6.67 + 3.85i)23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (0.223 − 0.387i)5-s + (0.154 − 0.987i)7-s − 0.353i·8-s + (0.273 + 0.158i)10-s + (−1.03 + 0.596i)11-s + (0.784 − 0.452i)13-s + (0.698 + 0.109i)14-s + 0.250·16-s + (0.120 − 0.208i)17-s + (−1.12 + 0.650i)19-s + (−0.111 + 0.193i)20-s + (−0.421 − 0.730i)22-s + (1.39 + 0.803i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.578 + 0.815i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.578 + 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1890\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $0.578 + 0.815i$
Analytic conductor: \(15.0917\)
Root analytic conductor: \(3.88480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1890} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1890,\ (\ :1/2),\ 0.578 + 0.815i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.294180289\)
\(L(\frac12)\) \(\approx\) \(1.294180289\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-0.408 + 2.61i)T \)
good11 \( 1 + (3.42 - 1.97i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.82 + 1.63i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.497 + 0.861i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.90 - 2.83i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-6.67 - 3.85i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.10 - 2.36i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.58iT - 31T^{2} \)
37 \( 1 + (5.05 + 8.75i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.16 + 8.94i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.10 + 8.83i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 4.73T + 47T^{2} \)
53 \( 1 + (8.82 + 5.09i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 3.40T + 59T^{2} \)
61 \( 1 + 6.29iT - 61T^{2} \)
67 \( 1 - 9.22T + 67T^{2} \)
71 \( 1 - 2.74iT - 71T^{2} \)
73 \( 1 + (12.7 + 7.36i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 1.84T + 79T^{2} \)
83 \( 1 + (-0.789 + 1.36i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (6.92 + 11.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.776 + 0.448i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.890955996693384056226455352492, −8.269736181716534627496377309438, −7.42739608921937603850029609500, −6.94021222972456623280758056284, −5.79176518377851858559223476310, −5.18836137936064103247975697477, −4.28445582106600875343352749607, −3.42496556518181522522837203117, −1.90576129749948838331810147023, −0.49551760071187834012248831367, 1.34572168416461133480153417023, 2.67170731824305298878023349326, 3.00955396525165790987376909458, 4.45772693824674139465044167718, 5.16702367278566804283692479890, 6.15268772289305647900578465152, 6.77703045117307789256410442779, 8.250535246090501379819129332316, 8.508265172754118726700131482513, 9.319153878798186306919667504873

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