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Properties

Label 2-1890-63.5-c1-0-25
Degree $2$
Conductor $1890$
Sign $0.609 + 0.792i$
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  + i·2-s − 4-s + (0.5 + 0.866i)5-s + (0.281 + 2.63i)7-s − i·8-s + (−0.866 + 0.5i)10-s + (−0.390 − 0.225i)11-s + (−4.26 − 2.46i)13-s + (−2.63 + 0.281i)14-s + 16-s + (−3.93 − 6.81i)17-s + (−4.75 − 2.74i)19-s + (−0.5 − 0.866i)20-s + (0.225 − 0.390i)22-s + (4.21 − 2.43i)23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (0.223 + 0.387i)5-s + (0.106 + 0.994i)7-s − 0.353i·8-s + (−0.273 + 0.158i)10-s + (−0.117 − 0.0679i)11-s + (−1.18 − 0.682i)13-s + (−0.703 + 0.0753i)14-s + 0.250·16-s + (−0.953 − 1.65i)17-s + (−1.08 − 0.629i)19-s + (−0.111 − 0.193i)20-s + (0.0480 − 0.0831i)22-s + (0.878 − 0.506i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.609 + 0.792i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.609 + 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1890\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $0.609 + 0.792i$
Analytic conductor: \(15.0917\)
Root analytic conductor: \(3.88480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1890} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1890,\ (\ :1/2),\ 0.609 + 0.792i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7768732468\)
\(L(\frac12)\) \(\approx\) \(0.7768732468\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-0.281 - 2.63i)T \)
good11 \( 1 + (0.390 + 0.225i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.26 + 2.46i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.93 + 6.81i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.75 + 2.74i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.21 + 2.43i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-8.05 + 4.65i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.574iT - 31T^{2} \)
37 \( 1 + (0.721 - 1.25i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.956 + 1.65i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.459 + 0.795i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 7.42T + 47T^{2} \)
53 \( 1 + (-8.30 + 4.79i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 8.86T + 59T^{2} \)
61 \( 1 - 9.86iT - 61T^{2} \)
67 \( 1 + 4.64T + 67T^{2} \)
71 \( 1 - 3.88iT - 71T^{2} \)
73 \( 1 + (-4.91 + 2.83i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 2.00T + 79T^{2} \)
83 \( 1 + (4.76 + 8.25i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.98 - 3.43i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-8.69 + 5.01i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.946404830586825854110954307211, −8.408019474532743098212008827716, −7.38050334580185746763866612809, −6.76515728423432343865098176496, −6.00793998393085119639576855163, −4.97687196527427857402509139855, −4.63185743527164018420413792075, −2.87591998962012237009113359798, −2.41774429236919522217002895533, −0.28254343147318177158615062435, 1.34797424300737827064279296686, 2.22379686832554505969928605566, 3.52354661503013019428186892474, 4.43912466453367244829302294575, 4.88659473402092345944141886565, 6.22429415326131110579859583555, 6.93256426651009138636022428622, 7.950020078873528809252069156772, 8.656956471108050838207636041836, 9.408964985376429783216067631011

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