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Properties

Label 2-1890-105.104-c1-0-39
Degree $2$
Conductor $1890$
Sign $0.979 + 0.202i$
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  − 2-s + 4-s + (1.73 + 1.41i)5-s + (2.34 − 1.22i)7-s − 8-s + (−1.73 − 1.41i)10-s − 5.76i·11-s + 6.42·13-s + (−2.34 + 1.22i)14-s + 16-s + 4.33i·17-s + 4.24i·19-s + (1.73 + 1.41i)20-s + 5.76i·22-s + 3·23-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + (0.774 + 0.632i)5-s + (0.886 − 0.462i)7-s − 0.353·8-s + (−0.547 − 0.447i)10-s − 1.73i·11-s + 1.78·13-s + (−0.626 + 0.327i)14-s + 0.250·16-s + 1.05i·17-s + 0.973i·19-s + (0.387 + 0.316i)20-s + 1.22i·22-s + 0.625·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.202i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 + 0.202i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1890\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $0.979 + 0.202i$
Analytic conductor: \(15.0917\)
Root analytic conductor: \(3.88480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1890} (1889, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1890,\ (\ :1/2),\ 0.979 + 0.202i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.806244027\)
\(L(\frac12)\) \(\approx\) \(1.806244027\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 + (-1.73 - 1.41i)T \)
7 \( 1 + (-2.34 + 1.22i)T \)
good11 \( 1 + 5.76iT - 11T^{2} \)
13 \( 1 - 6.42T + 13T^{2} \)
17 \( 1 - 4.33iT - 17T^{2} \)
19 \( 1 - 4.24iT - 19T^{2} \)
23 \( 1 - 3T + 23T^{2} \)
29 \( 1 + 3.31iT - 29T^{2} \)
31 \( 1 + 9.98iT - 31T^{2} \)
37 \( 1 - 4.89iT - 37T^{2} \)
41 \( 1 + 3.46T + 41T^{2} \)
43 \( 1 + 9.94iT - 43T^{2} \)
47 \( 1 + 2.91iT - 47T^{2} \)
53 \( 1 + 14.1T + 53T^{2} \)
59 \( 1 + 3.46T + 59T^{2} \)
61 \( 1 - 4.24iT - 61T^{2} \)
67 \( 1 - 4.89iT - 67T^{2} \)
71 \( 1 - 6.63iT - 71T^{2} \)
73 \( 1 - 4.69T + 73T^{2} \)
79 \( 1 - 7.12T + 79T^{2} \)
83 \( 1 - 7.07iT - 83T^{2} \)
89 \( 1 + 3.46T + 89T^{2} \)
97 \( 1 + 2.23T + 97T^{2} \)
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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

−9.074648799907203692530150673656, −8.278147025942117976171265575928, −8.023086706179366633282918845406, −6.74627583049316341993552671362, −6.00887105335998772282876131119, −5.62309770838019406880258981421, −3.94933172654051677349769552157, −3.26991543676884477140664589957, −1.89874828819178268360211499674, −1.02994202125719009871582792302, 1.23265791184777829758519404117, 1.88761508943595092559834730532, 3.05856333981129784656629942748, 4.72955700917338688526032675445, 5.00440582404398482287630994900, 6.20030218707488457683272795384, 6.93707335057231199296616737044, 7.81719927454059470155983730780, 8.703860056158490710940263029783, 9.155530430594430857679827800629

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