Properties

Label 2-1890-63.59-c1-0-1
Degree $2$
Conductor $1890$
Sign $-0.901 + 0.432i$
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 + (−1.80 + 1.93i)7-s + 0.999i·8-s + (−0.866 − 0.5i)10-s + 0.781i·11-s + (−2.26 − 1.31i)13-s + (−2.52 + 0.774i)14-s + (−0.5 + 0.866i)16-s + (−3.02 + 5.24i)17-s + (6.09 − 3.51i)19-s + (−0.499 − 0.866i)20-s + (−0.390 + 0.676i)22-s − 4.60i·23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s − 0.447·5-s + (−0.681 + 0.731i)7-s + 0.353i·8-s + (−0.273 − 0.158i)10-s + 0.235i·11-s + (−0.629 − 0.363i)13-s + (−0.676 + 0.206i)14-s + (−0.125 + 0.216i)16-s + (−0.734 + 1.27i)17-s + (1.39 − 0.807i)19-s + (−0.111 − 0.193i)20-s + (−0.0833 + 0.144i)22-s − 0.961i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.901 + 0.432i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.901 + 0.432i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1890\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $-0.901 + 0.432i$
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: \((2,\ 1890,\ (\ :1/2),\ -0.901 + 0.432i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4464495822\)
\(L(\frac12)\) \(\approx\) \(0.4464495822\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.80 - 1.93i)T \)
good11 \( 1 - 0.781iT - 11T^{2} \)
13 \( 1 + (2.26 + 1.31i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.02 - 5.24i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-6.09 + 3.51i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 4.60iT - 23T^{2} \)
29 \( 1 + (3.96 - 2.28i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (6.47 - 3.74i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.42 + 9.40i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.46 - 9.47i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.45 + 7.70i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.501 - 0.868i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (8.30 + 4.79i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.32 + 2.28i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.84 - 1.06i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.02 + 10.4i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.1iT - 71T^{2} \)
73 \( 1 + (-7.67 - 4.42i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.96 - 13.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.03 - 3.52i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-3.78 - 6.56i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.00 + 2.88i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.505404302824175315412918620417, −8.820668859456286611735704329425, −8.043968809157481025795233412101, −7.07798776494073977458804979491, −6.60985503119087521252506441261, −5.50846830049104801003189293575, −4.98678616830148037007257812793, −3.81761903534443469835338618441, −3.09809741900364177095845435144, −2.01748381620880200144222314449, 0.12320317329289372879673555147, 1.64041976289548569261704914606, 3.06788857042739542231911463694, 3.59614903389791767291341585033, 4.60622764758792628123187880075, 5.37234745506030828776669827301, 6.36699785296681159060584315048, 7.31191749232282369509703885350, 7.59075628167219080709665215154, 9.030568661341523621519042609696

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