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Properties

Label 2-1890-63.59-c1-0-16
Degree $2$
Conductor $1890$
Sign $0.916 + 0.400i$
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  + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + 5-s + (0.5 − 2.59i)7-s − 0.999i·8-s + (−0.866 − 0.5i)10-s + 1.26i·11-s + (3 + 1.73i)13-s + (−1.73 + 2i)14-s + (−0.5 + 0.866i)16-s + (4.09 − 2.36i)19-s + (0.499 + 0.866i)20-s + (0.633 − 1.09i)22-s + 9.46i·23-s + 25-s + (−1.73 − 3i)26-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + 0.447·5-s + (0.188 − 0.981i)7-s − 0.353i·8-s + (−0.273 − 0.158i)10-s + 0.382i·11-s + (0.832 + 0.480i)13-s + (−0.462 + 0.534i)14-s + (−0.125 + 0.216i)16-s + (0.940 − 0.542i)19-s + (0.111 + 0.193i)20-s + (0.135 − 0.234i)22-s + 1.97i·23-s + 0.200·25-s + (−0.339 − 0.588i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 + 0.400i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.916 + 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1890\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $0.916 + 0.400i$
Analytic conductor: \(15.0917\)
Root analytic conductor: \(3.88480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1890} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1890,\ (\ :1/2),\ 0.916 + 0.400i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.536823132\)
\(L(\frac12)\) \(\approx\) \(1.536823132\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 - T \)
7 \( 1 + (-0.5 + 2.59i)T \)
good11 \( 1 - 1.26iT - 11T^{2} \)
13 \( 1 + (-3 - 1.73i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.09 + 2.36i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 9.46iT - 23T^{2} \)
29 \( 1 + (-0.401 + 0.232i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.90 - 1.09i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.09 - 3.63i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.5 + 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.59 - 6.23i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.5 + 7.79i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (9.29 + 5.36i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.09 - 7.09i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.803 - 0.464i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.26iT - 71T^{2} \)
73 \( 1 + (-6 - 3.46i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-8.29 + 14.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.401 + 0.696i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (8.19 + 14.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (13.3 - 7.73i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.382549511104471690567499215833, −8.492472177540914787134751595022, −7.49791938045666595097146062625, −7.12162950310833797032490975475, −6.07936311725552066158346732860, −5.09121769640785305560350612320, −4.01702095327715161981027986610, −3.23528949809443533123803202199, −1.88092170638832467955882764172, −0.997989525314815771740733468066, 0.937104228195237914368572305109, 2.21318352864690108270735687474, 3.13454606876481071809632455179, 4.51697047447709702669752078414, 5.63186547032718933548113572620, 5.97725758746053599472613173729, 6.85643125017405143411578087598, 8.035637678905091419212237372564, 8.373015671634731548109827943675, 9.277549827791396640361436587252

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