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Properties

Label 2-855-19.5-c1-0-31
Degree $2$
Conductor $855$
Sign $0.124 + 0.992i$
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  + (2.42 − 0.883i)2-s + (3.57 − 3.00i)4-s + (−0.766 − 0.642i)5-s + (0.200 + 0.347i)7-s + (3.45 − 5.97i)8-s + (−2.42 − 0.883i)10-s + (2.59 − 4.49i)11-s + (0.501 + 2.84i)13-s + (0.794 + 0.666i)14-s + (1.47 − 8.35i)16-s + (−3.89 + 1.41i)17-s + (0.386 − 4.34i)19-s − 4.67·20-s + (2.32 − 13.2i)22-s + (−2.57 + 2.15i)23-s + ⋯
L(s)  = 1  + (1.71 − 0.624i)2-s + (1.78 − 1.50i)4-s + (−0.342 − 0.287i)5-s + (0.0759 + 0.131i)7-s + (1.22 − 2.11i)8-s + (−0.767 − 0.279i)10-s + (0.782 − 1.35i)11-s + (0.139 + 0.789i)13-s + (0.212 + 0.178i)14-s + (0.368 − 2.08i)16-s + (−0.944 + 0.343i)17-s + (0.0887 − 0.996i)19-s − 1.04·20-s + (0.496 − 2.81i)22-s + (−0.536 + 0.450i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.124 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.124 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $0.124 + 0.992i$
Analytic conductor: \(6.82720\)
Root analytic conductor: \(2.61289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :1/2),\ 0.124 + 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.07026 - 2.70949i\)
\(L(\frac12)\) \(\approx\) \(3.07026 - 2.70949i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.766 + 0.642i)T \)
19 \( 1 + (-0.386 + 4.34i)T \)
good2 \( 1 + (-2.42 + 0.883i)T + (1.53 - 1.28i)T^{2} \)
7 \( 1 + (-0.200 - 0.347i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.59 + 4.49i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.501 - 2.84i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (3.89 - 1.41i)T + (13.0 - 10.9i)T^{2} \)
23 \( 1 + (2.57 - 2.15i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (-6.18 - 2.25i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (-3.13 - 5.42i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 1.14T + 37T^{2} \)
41 \( 1 + (0.496 - 2.81i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (9.52 + 7.99i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (-6.35 - 2.31i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (9.42 - 7.90i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (1.42 - 0.518i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (5.35 - 4.49i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-0.711 - 0.258i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-6.38 - 5.35i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (1.72 - 9.76i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (-0.553 + 3.13i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-2.75 - 4.77i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.17 - 12.3i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (-8.35 + 3.04i)T + (74.3 - 62.3i)T^{2} \)
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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

−10.48174616109737058287022610118, −9.141021386977141248871326148784, −8.419826241566808825900255186361, −6.83601615198630574297860821414, −6.35908957481828464322369386430, −5.31242392479596073284139466069, −4.44288969418826721996029698395, −3.69396026210523645721113645394, −2.70569512140359198929976372297, −1.33744992007740213584148215829, 2.14695221605922255746829405443, 3.31529953013953636867198139585, 4.30579204854178061353813634965, 4.80041886241048404891564089099, 6.11881520438410494647179127883, 6.60815808593830592025320951372, 7.55523950357230214497755979119, 8.181082013145177416256754445171, 9.644787780883480730137418101373, 10.60422948572606093565408253776

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