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Properties

Label 2-855-171.49-c1-0-3
Degree $2$
Conductor $855$
Sign $-0.575 - 0.817i$
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  − 2.16·2-s + (−1.36 + 1.06i)3-s + 2.67·4-s + (0.5 + 0.866i)5-s + (2.94 − 2.31i)6-s + (−2.59 − 4.50i)7-s − 1.46·8-s + (0.717 − 2.91i)9-s + (−1.08 − 1.87i)10-s + (−1.09 − 1.89i)11-s + (−3.65 + 2.86i)12-s − 5.51·13-s + (5.62 + 9.73i)14-s + (−1.60 − 0.646i)15-s − 2.18·16-s + (2.29 − 3.98i)17-s + ⋯
L(s)  = 1  − 1.52·2-s + (−0.787 + 0.616i)3-s + 1.33·4-s + (0.223 + 0.387i)5-s + (1.20 − 0.943i)6-s + (−0.982 − 1.70i)7-s − 0.519·8-s + (0.239 − 0.970i)9-s + (−0.342 − 0.592i)10-s + (−0.329 − 0.570i)11-s + (−1.05 + 0.826i)12-s − 1.52·13-s + (1.50 + 2.60i)14-s + (−0.414 − 0.166i)15-s − 0.545·16-s + (0.557 − 0.965i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.575 - 0.817i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.575 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $-0.575 - 0.817i$
Analytic conductor: \(6.82720\)
Root analytic conductor: \(2.61289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :1/2),\ -0.575 - 0.817i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0471962 + 0.0909396i\)
\(L(\frac12)\) \(\approx\) \(0.0471962 + 0.0909396i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.36 - 1.06i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-1.96 - 3.89i)T \)
good2 \( 1 + 2.16T + 2T^{2} \)
7 \( 1 + (2.59 + 4.50i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.09 + 1.89i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 5.51T + 13T^{2} \)
17 \( 1 + (-2.29 + 3.98i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + 0.0138T + 23T^{2} \)
29 \( 1 + (0.758 - 1.31i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.58 + 4.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 10.6T + 37T^{2} \)
41 \( 1 + (-1.90 - 3.30i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 4.11T + 43T^{2} \)
47 \( 1 + (-0.222 + 0.384i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.54 - 9.61i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.24 + 2.14i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.23 + 3.86i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 - 4.85T + 67T^{2} \)
71 \( 1 + (0.669 - 1.16i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.38 - 9.31i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 3.20T + 79T^{2} \)
83 \( 1 + (-0.202 - 0.350i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.34 - 4.06i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 11.3T + 97T^{2} \)
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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

−10.17959609602107380503056818173, −9.880925619397740050144792023063, −9.233425953350698836411556867166, −7.74633874394073924802210871718, −7.22947443071872960425399064043, −6.53008627882498462748371792022, −5.31726449471940192855980915429, −4.08193348670460876130530246717, −2.93159238422722784405958659633, −0.923194619972578420035496260548, 0.11825123762738787744301805607, 1.85039293127420180365258554751, 2.64779976272699471784199922808, 4.95425549458551213763466627153, 5.65653394832436723254681634304, 6.71456383365207713533084845329, 7.34930808431953388986337781976, 8.371111107178535204867093139182, 9.059595589491619473987394266774, 9.886126858481622875370487269751

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