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Properties

Label 2-1890-315.299-c1-0-12
Degree $2$
Conductor $1890$
Sign $-0.600 - 0.799i$
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  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (2.05 − 0.889i)5-s + (0.970 + 2.46i)7-s − 0.999·8-s + (1.79 + 1.33i)10-s + 2.28i·11-s + (2.02 + 3.49i)13-s + (−1.64 + 2.07i)14-s + (−0.5 − 0.866i)16-s + (−5.58 + 3.22i)17-s + (−0.177 − 0.102i)19-s + (−0.255 + 2.22i)20-s + (−1.98 + 1.14i)22-s − 2.00·23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.917 − 0.397i)5-s + (0.366 + 0.930i)7-s − 0.353·8-s + (0.567 + 0.421i)10-s + 0.689i·11-s + (0.560 + 0.970i)13-s + (−0.439 + 0.553i)14-s + (−0.125 − 0.216i)16-s + (−1.35 + 0.782i)17-s + (−0.0407 − 0.0235i)19-s + (−0.0572 + 0.496i)20-s + (−0.422 + 0.243i)22-s − 0.418·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.600 - 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.600 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1890\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $-0.600 - 0.799i$
Analytic conductor: \(15.0917\)
Root analytic conductor: \(3.88480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1890} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1890,\ (\ :1/2),\ -0.600 - 0.799i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.217683275\)
\(L(\frac12)\) \(\approx\) \(2.217683275\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 \)
5 \( 1 + (-2.05 + 0.889i)T \)
7 \( 1 + (-0.970 - 2.46i)T \)
good11 \( 1 - 2.28iT - 11T^{2} \)
13 \( 1 + (-2.02 - 3.49i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (5.58 - 3.22i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.177 + 0.102i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 2.00T + 23T^{2} \)
29 \( 1 + (2.95 + 1.70i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.844 + 0.487i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-7.40 - 4.27i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.15 + 2.00i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.12 + 1.22i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.13 - 2.38i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.35 - 4.08i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.53 + 7.85i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-9.33 + 5.39i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.24 + 3.02i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 11.0iT - 71T^{2} \)
73 \( 1 + (-7.78 - 13.4i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.61 - 6.26i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (9.91 + 5.72i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-5.08 + 8.81i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.24 - 7.34i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.317555155700915865925024007842, −8.672095902039021556896793716350, −8.093713483001010644835139571087, −6.82647891746804212678219728455, −6.30306416807491780671560002992, −5.55998227699694814049353758167, −4.70970328959369117755683848694, −4.01419814131871632741910453619, −2.42582282296280102068705338507, −1.74031600027041299360093238019, 0.69960487337955385312250281207, 1.93193660672603627770485381003, 2.94633909342826653354997487012, 3.82411611724576511721409041627, 4.82942299366562545848330272187, 5.66624388100748107882898100963, 6.42532486260136752581099983952, 7.26996327892094416781145577991, 8.275016358147726109978328395169, 9.106953830567870120080570406822

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