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Properties

Label 2-1890-315.104-c1-0-25
Degree $2$
Conductor $1890$
Sign $0.999 + 0.0203i$
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 + (−0.797 + 2.08i)5-s + (0.843 + 2.50i)7-s + 0.999·8-s + (2.20 − 0.353i)10-s + (4.88 − 2.82i)11-s + (0.902 − 1.56i)13-s + (1.74 − 1.98i)14-s + (−0.5 − 0.866i)16-s − 4.43i·17-s + 4.20i·19-s + (−1.41 − 1.73i)20-s + (−4.88 − 2.82i)22-s + (3.13 − 5.43i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.356 + 0.934i)5-s + (0.318 + 0.947i)7-s + 0.353·8-s + (0.698 − 0.111i)10-s + (1.47 − 0.850i)11-s + (0.250 − 0.433i)13-s + (0.467 − 0.530i)14-s + (−0.125 − 0.216i)16-s − 1.07i·17-s + 0.964i·19-s + (−0.315 − 0.388i)20-s + (−1.04 − 0.601i)22-s + (0.653 − 1.13i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0203i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0203i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1890\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $0.999 + 0.0203i$
Analytic conductor: \(15.0917\)
Root analytic conductor: \(3.88480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1890} (1259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1890,\ (\ :1/2),\ 0.999 + 0.0203i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.528831428\)
\(L(\frac12)\) \(\approx\) \(1.528831428\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.797 - 2.08i)T \)
7 \( 1 + (-0.843 - 2.50i)T \)
good11 \( 1 + (-4.88 + 2.82i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.902 + 1.56i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 4.43iT - 17T^{2} \)
19 \( 1 - 4.20iT - 19T^{2} \)
23 \( 1 + (-3.13 + 5.43i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.728 + 0.420i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.34 - 1.35i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 3.95iT - 37T^{2} \)
41 \( 1 + (5.47 - 9.48i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-10.1 + 5.88i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.36 + 1.94i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 0.269T + 53T^{2} \)
59 \( 1 + (-2.74 + 4.74i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.75 + 3.90i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.51 + 2.02i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.50iT - 71T^{2} \)
73 \( 1 - 11.8T + 73T^{2} \)
79 \( 1 + (-3.09 - 5.36i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.53 + 3.77i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 6.72T + 89T^{2} \)
97 \( 1 + (-7.17 - 12.4i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.206515635292994706079815759527, −8.510358445967902033329765700991, −7.911568511475871029991251407877, −6.76757388205174109067690859608, −6.21850724620377317453147340688, −5.11702771376688096863725904550, −3.95806871960840928685861042286, −3.16687314153416201496136375167, −2.37402915940749632925811432541, −0.969086676571981933951400162422, 0.913412105523701971507984030015, 1.74005536187584464215758159396, 3.84051765143827292519826018903, 4.25327824473771699858680350704, 5.12386087382551166706580087508, 6.18355997341328180257357746562, 7.07435326756336510436704751725, 7.51942755794076747293883700420, 8.520600185108807257277429060931, 9.127016837802740934876308403396

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