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Properties

Label 2-1860-465.464-c1-0-12
Degree $2$
Conductor $1860$
Sign $0.655 - 0.755i$
Analytic cond. $14.8521$
Root an. cond. $3.85385$
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  + (−1.30 − 1.13i)3-s − 2.23i·5-s + (0.423 + 2.96i)9-s + 6.94·13-s + (−2.53 + 2.92i)15-s + 8.11i·17-s − 8.59·19-s + 4.87i·23-s − 5.00·25-s + (2.81 − 4.36i)27-s − 5.56·31-s − 11.6·37-s + (−9.09 − 7.88i)39-s + 8.82i·41-s + 8.02·43-s + ⋯
L(s)  = 1  + (−0.755 − 0.655i)3-s − 0.999i·5-s + (0.141 + 0.989i)9-s + 1.92·13-s + (−0.655 + 0.755i)15-s + 1.96i·17-s − 1.97·19-s + 1.01i·23-s − 1.00·25-s + (0.542 − 0.840i)27-s − 1.00·31-s − 1.91·37-s + (−1.45 − 1.26i)39-s + 1.37i·41-s + 1.22·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.655 - 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.655 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1860\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.655 - 0.755i$
Analytic conductor: \(14.8521\)
Root analytic conductor: \(3.85385\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1860} (929, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1860,\ (\ :1/2),\ 0.655 - 0.755i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8780832504\)
\(L(\frac12)\) \(\approx\) \(0.8780832504\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.30 + 1.13i)T \)
5 \( 1 + 2.23iT \)
31 \( 1 + 5.56T \)
good7 \( 1 - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 6.94T + 13T^{2} \)
17 \( 1 - 8.11iT - 17T^{2} \)
19 \( 1 + 8.59T + 19T^{2} \)
23 \( 1 - 4.87iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
37 \( 1 + 11.6T + 37T^{2} \)
41 \( 1 - 8.82iT - 41T^{2} \)
43 \( 1 - 8.02T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 12.3iT - 53T^{2} \)
59 \( 1 + 1.56iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 13.3iT - 71T^{2} \)
73 \( 1 - 2.44T + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 - 1.32iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−9.046364861914213633052615143063, −8.485167998297196477881463687723, −7.968381056527428116553567010144, −6.80653735842218306181011943184, −5.92189643253208979854008176221, −5.70710085606117563311964591676, −4.35846390956855490831417612878, −3.77453430527922728324779747125, −1.87768470958076940415396520188, −1.26509262306282174211935546232, 0.38139762328882492004613827073, 2.18069929235592057839426195137, 3.43896908724563219362423212558, 4.04606237687060312092660073720, 5.12349849816148679930532690396, 6.03075290760343983900957271120, 6.61807993186470968661769997077, 7.28128950257846491085654792632, 8.642405856250692428120389526277, 9.052331009197492355467670681441

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