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Properties

Label 2-855-19.17-c1-0-32
Degree $2$
Conductor $855$
Sign $-0.960 - 0.278i$
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  + (0.253 − 1.44i)2-s + (−0.130 − 0.0473i)4-s + (−0.939 + 0.342i)5-s + (−2.03 − 3.52i)7-s + (1.36 − 2.35i)8-s + (0.253 + 1.44i)10-s + (−0.310 + 0.537i)11-s + (−3.90 + 3.27i)13-s + (−5.59 + 2.03i)14-s + (−3.26 − 2.73i)16-s + (0.0462 − 0.262i)17-s + (0.399 − 4.34i)19-s + 0.138·20-s + (0.695 + 0.583i)22-s + (−5.48 − 1.99i)23-s + ⋯
L(s)  = 1  + (0.179 − 1.01i)2-s + (−0.0650 − 0.0236i)4-s + (−0.420 + 0.152i)5-s + (−0.769 − 1.33i)7-s + (0.481 − 0.833i)8-s + (0.0802 + 0.455i)10-s + (−0.0936 + 0.162i)11-s + (−1.08 + 0.908i)13-s + (−1.49 + 0.544i)14-s + (−0.815 − 0.684i)16-s + (0.0112 − 0.0636i)17-s + (0.0917 − 0.995i)19-s + 0.0309·20-s + (0.148 + 0.124i)22-s + (−1.14 − 0.416i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 - 0.278i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.960 - 0.278i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $-0.960 - 0.278i$
Analytic conductor: \(6.82720\)
Root analytic conductor: \(2.61289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :1/2),\ -0.960 - 0.278i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.132590 + 0.933364i\)
\(L(\frac12)\) \(\approx\) \(0.132590 + 0.933364i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.939 - 0.342i)T \)
19 \( 1 + (-0.399 + 4.34i)T \)
good2 \( 1 + (-0.253 + 1.44i)T + (-1.87 - 0.684i)T^{2} \)
7 \( 1 + (2.03 + 3.52i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.310 - 0.537i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.90 - 3.27i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-0.0462 + 0.262i)T + (-15.9 - 5.81i)T^{2} \)
23 \( 1 + (5.48 + 1.99i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (0.708 + 4.01i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (-3.24 - 5.62i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 8.83T + 37T^{2} \)
41 \( 1 + (-3.43 - 2.88i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (-1.69 + 0.615i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (2.00 + 11.3i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + (2.37 + 0.862i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (0.154 - 0.876i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (2.03 + 0.742i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (2.44 + 13.8i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (1.54 - 0.563i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (-1.37 - 1.15i)T + (12.6 + 71.8i)T^{2} \)
79 \( 1 + (-3.94 - 3.30i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-6.89 - 11.9i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.000572 + 0.000480i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (1.80 - 10.2i)T + (-91.1 - 33.1i)T^{2} \)
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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

−10.05400310883163219755534947886, −9.261672355669199292048076777514, −7.86412566821539197274733236762, −6.98740157726695341490994258374, −6.64524245240975081807501250922, −4.77288077982781387260565973382, −4.04867678771926116749962573868, −3.18048940255799940857460915656, −2.05541797026759130019245814013, −0.39669686313096904331764395343, 2.17043271663824419284206803104, 3.27321832557500147001034224671, 4.73438140428339983065992547414, 5.74606989650517085665362073216, 6.03694927534092159768484222630, 7.29412033298037884104163915587, 7.919908933845696405761055749506, 8.707935483901401853804079828523, 9.688177404613218361458163970640, 10.49862070842311967449327127584

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