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Properties

Label 2-1890-63.16-c1-0-25
Degree $2$
Conductor $1890$
Sign $0.754 + 0.656i$
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 − 5-s + (2.32 + 1.26i)7-s − 0.999·8-s + (−0.5 − 0.866i)10-s − 0.961·11-s + (−2.94 − 5.09i)13-s + (0.0665 + 2.64i)14-s + (−0.5 − 0.866i)16-s + (−3.89 − 6.75i)17-s + (0.774 − 1.34i)19-s + (0.499 − 0.866i)20-s + (−0.480 − 0.832i)22-s + 3.90·23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s − 0.447·5-s + (0.878 + 0.478i)7-s − 0.353·8-s + (−0.158 − 0.273i)10-s − 0.289·11-s + (−0.816 − 1.41i)13-s + (0.0177 + 0.706i)14-s + (−0.125 − 0.216i)16-s + (−0.945 − 1.63i)17-s + (0.177 − 0.307i)19-s + (0.111 − 0.193i)20-s + (−0.102 − 0.177i)22-s + 0.815·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.754 + 0.656i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.754 + 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1890\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $0.754 + 0.656i$
Analytic conductor: \(15.0917\)
Root analytic conductor: \(3.88480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1890} (1801, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1890,\ (\ :1/2),\ 0.754 + 0.656i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.391622671\)
\(L(\frac12)\) \(\approx\) \(1.391622671\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 + (-2.32 - 1.26i)T \)
good11 \( 1 + 0.961T + 11T^{2} \)
13 \( 1 + (2.94 + 5.09i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.89 + 6.75i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.774 + 1.34i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 3.90T + 23T^{2} \)
29 \( 1 + (-0.543 + 0.940i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.27 - 2.21i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.84 + 10.1i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.15 + 1.99i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.18 - 3.79i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.77 + 3.07i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.274 + 0.475i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.00 - 5.20i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.34 + 12.7i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.410 + 0.711i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.96T + 71T^{2} \)
73 \( 1 + (-0.368 - 0.638i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.39 - 5.88i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.12 - 10.6i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.85 + 4.94i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.80 + 13.5i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.981882746879150065330032528199, −8.160509927460635258008173200037, −7.51325992570934193948310177023, −6.97387116983688033536335706413, −5.72675939215627774863882562806, −5.02979752470569873845818519806, −4.57048569362549697462767316461, −3.16678488436135811323564926907, −2.40878878273409408631606139550, −0.45979266899177313800421769211, 1.38481979457883545043641951771, 2.27318220855669481529171510918, 3.55702059350772106892659820482, 4.48811510790736243258254866088, 4.79351766561970569821782840409, 6.11003779634562097476928498279, 6.92966546923034388791744660960, 7.81289565007324815393871232137, 8.565930204330562423301583882252, 9.339458690416303439148879108373

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