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Properties

Label 2-855-19.6-c1-0-20
Degree $2$
Conductor $855$
Sign $0.967 + 0.251i$
Analytic cond. $6.82720$
Root an. cond. $2.61289$
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  + (−1.39 + 1.16i)2-s + (0.226 − 1.28i)4-s + (0.173 + 0.984i)5-s + (−0.536 + 0.929i)7-s + (−0.634 − 1.09i)8-s + (−1.39 − 1.16i)10-s + (−1.65 − 2.86i)11-s + (−2.49 + 0.908i)13-s + (−0.338 − 1.92i)14-s + (4.61 + 1.67i)16-s + (3.06 − 2.57i)17-s + (0.281 − 4.34i)19-s + 1.30·20-s + (5.65 + 2.05i)22-s + (−0.304 + 1.72i)23-s + ⋯
L(s)  = 1  + (−0.984 + 0.825i)2-s + (0.113 − 0.640i)4-s + (0.0776 + 0.440i)5-s + (−0.202 + 0.351i)7-s + (−0.224 − 0.388i)8-s + (−0.440 − 0.369i)10-s + (−0.499 − 0.864i)11-s + (−0.692 + 0.252i)13-s + (−0.0905 − 0.513i)14-s + (1.15 + 0.419i)16-s + (0.743 − 0.623i)17-s + (0.0646 − 0.997i)19-s + 0.291·20-s + (1.20 + 0.438i)22-s + (−0.0635 + 0.360i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.251i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 + 0.251i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $0.967 + 0.251i$
Analytic conductor: \(6.82720\)
Root analytic conductor: \(2.61289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (766, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :1/2),\ 0.967 + 0.251i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.585397 - 0.0748694i\)
\(L(\frac12)\) \(\approx\) \(0.585397 - 0.0748694i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.173 - 0.984i)T \)
19 \( 1 + (-0.281 + 4.34i)T \)
good2 \( 1 + (1.39 - 1.16i)T + (0.347 - 1.96i)T^{2} \)
7 \( 1 + (0.536 - 0.929i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.65 + 2.86i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.49 - 0.908i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (-3.06 + 2.57i)T + (2.95 - 16.7i)T^{2} \)
23 \( 1 + (0.304 - 1.72i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (1.72 + 1.44i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-4.02 + 6.97i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 5.64T + 37T^{2} \)
41 \( 1 + (0.842 + 0.306i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (1.47 + 8.38i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-4.82 - 4.04i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + (-0.590 + 3.34i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (-1.13 + 0.955i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-2.38 + 13.5i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-9.85 - 8.26i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (1.91 + 10.8i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (-2.45 - 0.892i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (-6.17 - 2.24i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-1.34 + 2.33i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.742 + 0.270i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (14.3 - 12.0i)T + (16.8 - 95.5i)T^{2} \)
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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

−9.791513128048997889100337223854, −9.298281445958115203658113734387, −8.379638971445772804117979444153, −7.60829080126847128586749401261, −6.93082609054140883246782990276, −6.02546564190060611466061093806, −5.17107192437817094934238050671, −3.58148012611813358372320919339, −2.51752084933028556247997020141, −0.44399431133473672042708224998, 1.19895948700711136902203695832, 2.31076922357850794820406978372, 3.53163567206741501976277239064, 4.87258250118244373602536431857, 5.73352574853167952243474731463, 7.08027068941494750677454624991, 7.973266973009027145969307520312, 8.625620076089414470449188310008, 9.649579148126668515406481449623, 10.21665172970349281932567975690

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