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Properties

Label 2-1890-63.20-c1-0-18
Degree $2$
Conductor $1890$
Sign $0.833 + 0.553i$
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 + (−0.5 − 0.866i)5-s + (1.66 − 2.05i)7-s − 0.999i·8-s + 0.999i·10-s + (4.23 + 2.44i)11-s + (5.06 − 2.92i)13-s + (−2.47 + 0.943i)14-s + (−0.5 + 0.866i)16-s + 0.423·17-s + 6.07i·19-s + (0.499 − 0.866i)20-s + (−2.44 − 4.23i)22-s + (1.73 − 1.00i)23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.223 − 0.387i)5-s + (0.630 − 0.775i)7-s − 0.353i·8-s + 0.316i·10-s + (1.27 + 0.737i)11-s + (1.40 − 0.811i)13-s + (−0.660 + 0.252i)14-s + (−0.125 + 0.216i)16-s + 0.102·17-s + 1.39i·19-s + (0.111 − 0.193i)20-s + (−0.521 − 0.902i)22-s + (0.361 − 0.208i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.833 + 0.553i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.833 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1890\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $0.833 + 0.553i$
Analytic conductor: \(15.0917\)
Root analytic conductor: \(3.88480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1890} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1890,\ (\ :1/2),\ 0.833 + 0.553i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.589512462\)
\(L(\frac12)\) \(\approx\) \(1.589512462\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-1.66 + 2.05i)T \)
good11 \( 1 + (-4.23 - 2.44i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-5.06 + 2.92i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 0.423T + 17T^{2} \)
19 \( 1 - 6.07iT - 19T^{2} \)
23 \( 1 + (-1.73 + 1.00i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-8.22 - 4.74i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (7.78 - 4.49i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 1.70T + 37T^{2} \)
41 \( 1 + (1.90 + 3.29i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.237 - 0.411i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.424 + 0.736i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 8.33iT - 53T^{2} \)
59 \( 1 + (-2.06 - 3.57i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.0769 + 0.0444i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.41 - 7.64i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.82iT - 71T^{2} \)
73 \( 1 + 8.24iT - 73T^{2} \)
79 \( 1 + (-5.14 + 8.91i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.90 - 5.03i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 4.12T + 89T^{2} \)
97 \( 1 + (2.29 + 1.32i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.926652830189618699257023902481, −8.561184105746742602378960093046, −7.69944257433406788534643090549, −6.99145838152540237178691494004, −6.08749184275661137869449611109, −4.95179899634591828312452159395, −3.95380997853380481423572235233, −3.40386611673396642626897629910, −1.63650705861934530344411707948, −1.07570154858425552191193581740, 0.996581584720902697788773519327, 2.13844355068638762626659490923, 3.39070829152834069889678960121, 4.37301389977716187069397858034, 5.47368974394501000431518790115, 6.39188328032714817157281181091, 6.75225772710385486305642691490, 7.892499338576451494970759401789, 8.714995595258731965571474079626, 8.960534801958784664987771703871

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