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Properties

Label 2-855-19.17-c1-0-2
Degree $2$
Conductor $855$
Sign $0.285 + 0.958i$
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  + (−0.455 + 2.58i)2-s + (−4.58 − 1.66i)4-s + (−0.939 + 0.342i)5-s + (1.91 + 3.31i)7-s + (3.77 − 6.53i)8-s + (−0.455 − 2.58i)10-s + (−1.63 + 2.82i)11-s + (−0.278 + 0.234i)13-s + (−9.44 + 3.43i)14-s + (7.67 + 6.44i)16-s + (0.118 − 0.673i)17-s + (−3.86 + 2.01i)19-s + 4.87·20-s + (−6.56 − 5.50i)22-s + (−8.80 − 3.20i)23-s + ⋯
L(s)  = 1  + (−0.321 + 1.82i)2-s + (−2.29 − 0.833i)4-s + (−0.420 + 0.152i)5-s + (0.724 + 1.25i)7-s + (1.33 − 2.30i)8-s + (−0.143 − 0.816i)10-s + (−0.492 + 0.853i)11-s + (−0.0773 + 0.0649i)13-s + (−2.52 + 0.918i)14-s + (1.91 + 1.61i)16-s + (0.0288 − 0.163i)17-s + (−0.886 + 0.461i)19-s + 1.09·20-s + (−1.39 − 1.17i)22-s + (−1.83 − 0.668i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.285 + 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.285 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $0.285 + 0.958i$
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: \((2,\ 855,\ (\ :1/2),\ 0.285 + 0.958i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.363899 - 0.271232i\)
\(L(\frac12)\) \(\approx\) \(0.363899 - 0.271232i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.86 - 2.01i)T \)
good2 \( 1 + (0.455 - 2.58i)T + (-1.87 - 0.684i)T^{2} \)
7 \( 1 + (-1.91 - 3.31i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.63 - 2.82i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.278 - 0.234i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-0.118 + 0.673i)T + (-15.9 - 5.81i)T^{2} \)
23 \( 1 + (8.80 + 3.20i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (0.267 + 1.51i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (-1.22 - 2.12i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 0.163T + 37T^{2} \)
41 \( 1 + (-5.64 - 4.73i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (2.43 - 0.885i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (1.52 + 8.66i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + (-10.4 - 3.80i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (-1.35 + 7.67i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (2.27 + 0.827i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (1.65 + 9.41i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (4.81 - 1.75i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (6.03 + 5.06i)T + (12.6 + 71.8i)T^{2} \)
79 \( 1 + (7.14 + 5.99i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-0.364 - 0.631i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (9.68 - 8.12i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (1.31 - 7.43i)T + (-91.1 - 33.1i)T^{2} \)
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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

−10.47311747047902784256336143939, −9.703319914674128650135493329403, −8.721555855367979464289894132497, −8.174635295608637484970102823488, −7.59274844248535001975986828696, −6.53976876852106821619813375662, −5.81542457975565207465551953049, −4.94073104910152939765282713509, −4.17857545581913827829831419905, −2.18092997463280302658594759895, 0.25541922847722552325929263538, 1.47661795937231438469758176737, 2.74972418645641501336619087632, 4.00003658820009839524196084959, 4.30744526192305599563647603032, 5.64791112466247794660085210654, 7.36491763059247597532837615194, 8.165390654616559450128176175822, 8.741638519464879615729135237965, 9.926900947840572630887908508206

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