Properties

Label 2-855-95.14-c0-0-0
Degree $2$
Conductor $855$
Sign $-0.730 - 0.683i$
Analytic cond. $0.426700$
Root an. cond. $0.653223$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.20 + 0.439i)2-s + (0.500 − 0.419i)4-s + (−0.642 + 0.766i)5-s + (0.223 − 0.386i)8-s + (0.439 − 1.20i)10-s + (−0.213 + 1.20i)16-s + (0.642 + 1.76i)17-s + (−0.173 + 0.984i)19-s + 0.652i·20-s + (−0.642 − 0.766i)23-s + (−0.173 − 0.984i)25-s + (−1.70 + 0.984i)31-s + (−0.196 − 1.11i)32-s + (−1.55 − 1.85i)34-s + (−0.223 − 1.26i)38-s + ⋯
L(s)  = 1  + (−1.20 + 0.439i)2-s + (0.500 − 0.419i)4-s + (−0.642 + 0.766i)5-s + (0.223 − 0.386i)8-s + (0.439 − 1.20i)10-s + (−0.213 + 1.20i)16-s + (0.642 + 1.76i)17-s + (−0.173 + 0.984i)19-s + 0.652i·20-s + (−0.642 − 0.766i)23-s + (−0.173 − 0.984i)25-s + (−1.70 + 0.984i)31-s + (−0.196 − 1.11i)32-s + (−1.55 − 1.85i)34-s + (−0.223 − 1.26i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.730 - 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.730 - 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $-0.730 - 0.683i$
Analytic conductor: \(0.426700\)
Root analytic conductor: \(0.653223\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :0),\ -0.730 - 0.683i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3568392599\)
\(L(\frac12)\) \(\approx\) \(0.3568392599\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (0.642 - 0.766i)T \)
19 \( 1 + (0.173 - 0.984i)T \)
good2 \( 1 + (1.20 - 0.439i)T + (0.766 - 0.642i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.939 + 0.342i)T^{2} \)
17 \( 1 + (-0.642 - 1.76i)T + (-0.766 + 0.642i)T^{2} \)
23 \( 1 + (0.642 + 0.766i)T + (-0.173 + 0.984i)T^{2} \)
29 \( 1 + (-0.766 - 0.642i)T^{2} \)
31 \( 1 + (1.70 - 0.984i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (0.939 + 0.342i)T^{2} \)
43 \( 1 + (-0.173 - 0.984i)T^{2} \)
47 \( 1 + (0.524 - 1.43i)T + (-0.766 - 0.642i)T^{2} \)
53 \( 1 + (-0.524 + 0.439i)T + (0.173 - 0.984i)T^{2} \)
59 \( 1 + (-0.766 + 0.642i)T^{2} \)
61 \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \)
67 \( 1 + (0.766 + 0.642i)T^{2} \)
71 \( 1 + (-0.173 - 0.984i)T^{2} \)
73 \( 1 + (0.939 + 0.342i)T^{2} \)
79 \( 1 + (-1.70 - 0.300i)T + (0.939 + 0.342i)T^{2} \)
83 \( 1 + (0.300 - 0.173i)T + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.939 - 0.342i)T^{2} \)
97 \( 1 + (0.766 - 0.642i)T^{2} \)
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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

−10.51651262266626580927030374128, −9.917127207083053661727799834724, −8.831919183761697183655205736494, −8.083377054527619127392585143313, −7.61617575909176315098451239138, −6.61404559689627751773058512930, −5.90855075548525541528393027297, −4.21627728059452987189738723846, −3.42749585225663157951832239598, −1.69367753717295034609660843948, 0.54721432964599381798924581044, 2.02403356141327046263579741277, 3.43194313689422837073917706549, 4.76933945986132359492714720898, 5.47885630800582923062054540128, 7.15756646837573963870427366172, 7.68199587312803895347554861306, 8.566727721317085276744414658819, 9.334946251207563605357273489441, 9.734617961629327672429326547403

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