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Properties

Label 2-1890-315.209-c1-0-8
Degree $2$
Conductor $1890$
Sign $-0.929 - 0.368i$
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 + (−1.74 + 1.98i)7-s + 0.999·8-s + (−2.20 − 0.353i)10-s + (4.88 + 2.82i)11-s + (−0.902 − 1.56i)13-s + (−0.843 − 2.50i)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 + (−0.661 + 0.750i)7-s + 0.353·8-s + (−0.698 − 0.111i)10-s + (1.47 + 0.850i)11-s + (−0.250 − 0.433i)13-s + (−0.225 − 0.670i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 - 0.368i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1890\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $-0.929 - 0.368i$
Analytic conductor: \(15.0917\)
Root analytic conductor: \(3.88480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1890} (629, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1890,\ (\ :1/2),\ -0.929 - 0.368i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.267765727\)
\(L(\frac12)\) \(\approx\) \(1.267765727\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.74 - 1.98i)T \)
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

−9.426743442606163301187042742459, −9.111573119833445229310850327096, −7.72268315469002090890672155229, −7.22781867061370021236900486163, −6.37573790756160595905188655989, −5.90446490060291366771078222146, −4.88432819436201861474549763845, −3.66282740581668627823294504685, −2.73585284867199757739692955487, −1.51072289484717435408090160590, 0.56390838105384884124522507260, 1.46941106634061158451375539814, 2.79500585779773836830151586784, 4.02110063053525349393240261990, 4.35163340780628747064670708583, 5.73407254349597448980701492572, 6.51976714134546496217559152259, 7.30909602380280412456561251073, 8.500821821725456997774539385691, 9.000243038025964879451407957287

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