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Properties

Label 2-855-19.7-c1-0-31
Degree $2$
Conductor $855$
Sign $-0.813 - 0.582i$
Analytic cond. $6.82720$
Root an. cond. $2.61289$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − 1.73i)4-s + (−0.5 − 0.866i)5-s − 4·7-s − 3·11-s + (−1 + 1.73i)13-s + (−1.99 − 3.46i)16-s + (3 + 5.19i)17-s + (−3.5 + 2.59i)19-s − 1.99·20-s + (−0.499 + 0.866i)25-s + (−4 + 6.92i)28-s + (−1.5 + 2.59i)29-s − 7·31-s + (2 + 3.46i)35-s + 8·37-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)4-s + (−0.223 − 0.387i)5-s − 1.51·7-s − 0.904·11-s + (−0.277 + 0.480i)13-s + (−0.499 − 0.866i)16-s + (0.727 + 1.26i)17-s + (−0.802 + 0.596i)19-s − 0.447·20-s + (−0.0999 + 0.173i)25-s + (−0.755 + 1.30i)28-s + (−0.278 + 0.482i)29-s − 1.25·31-s + (0.338 + 0.585i)35-s + 1.31·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.813 - 0.582i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.813 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $-0.813 - 0.582i$
Analytic conductor: \(6.82720\)
Root analytic conductor: \(2.61289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (406, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 855,\ (\ :1/2),\ -0.813 - 0.582i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (3.5 - 2.59i)T \)
good2 \( 1 + (-1 + 1.73i)T^{2} \)
7 \( 1 + 4T + 7T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 7T + 31T^{2} \)
37 \( 1 - 8T + 37T^{2} \)
41 \( 1 + (3 + 5.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2 - 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (7.5 + 12.9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.5 - 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1 - 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.5 + 2.59i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (4 + 6.92i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.5 + 4.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + (7.5 - 12.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4 + 6.92i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.785501073806385524056213412322, −9.037732842181625587261620891294, −7.899332607614189981339704991996, −6.97271323239509172320534047471, −6.07952880579286187774227883498, −5.51980529134237890805706407274, −4.18387938804896131922298628918, −3.05477246548069317452286738383, −1.77095464773124816393736803881, 0, 2.66935415676807917451504337439, 3.01920831431247113420676509838, 4.17893331521981998392333728543, 5.58331007864135834733462007136, 6.55010406227323394050438693908, 7.32401323469590469361745876697, 7.88682278198065847964142894844, 9.072292990423787685858194721242, 9.850530987711527256674952408026

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