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Properties

Label 2-855-95.4-c1-0-16
Degree $2$
Conductor $855$
Sign $0.884 - 0.466i$
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  + (0.358 − 0.984i)2-s + (0.691 + 0.580i)4-s + (−0.296 + 2.21i)5-s + (2.37 + 1.37i)7-s + (2.63 − 1.52i)8-s + (2.07 + 1.08i)10-s + (0.416 + 0.721i)11-s + (−0.601 − 0.106i)13-s + (2.19 − 1.84i)14-s + (−0.239 − 1.35i)16-s + (−1.65 + 4.54i)17-s + (−4.35 − 0.175i)19-s + (−1.49 + 1.36i)20-s + (0.859 − 0.151i)22-s + (−2.41 + 2.87i)23-s + ⋯
L(s)  = 1  + (0.253 − 0.695i)2-s + (0.345 + 0.290i)4-s + (−0.132 + 0.991i)5-s + (0.896 + 0.517i)7-s + (0.930 − 0.537i)8-s + (0.656 + 0.343i)10-s + (0.125 + 0.217i)11-s + (−0.166 − 0.0294i)13-s + (0.587 − 0.493i)14-s + (−0.0598 − 0.339i)16-s + (−0.401 + 1.10i)17-s + (−0.999 − 0.0402i)19-s + (−0.333 + 0.304i)20-s + (0.183 − 0.0322i)22-s + (−0.502 + 0.599i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 - 0.466i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.884 - 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $0.884 - 0.466i$
Analytic conductor: \(6.82720\)
Root analytic conductor: \(2.61289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :1/2),\ 0.884 - 0.466i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.10031 + 0.520473i\)
\(L(\frac12)\) \(\approx\) \(2.10031 + 0.520473i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.296 - 2.21i)T \)
19 \( 1 + (4.35 + 0.175i)T \)
good2 \( 1 + (-0.358 + 0.984i)T + (-1.53 - 1.28i)T^{2} \)
7 \( 1 + (-2.37 - 1.37i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.416 - 0.721i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.601 + 0.106i)T + (12.2 + 4.44i)T^{2} \)
17 \( 1 + (1.65 - 4.54i)T + (-13.0 - 10.9i)T^{2} \)
23 \( 1 + (2.41 - 2.87i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (-3.73 + 1.35i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-3.46 + 5.99i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.33iT - 37T^{2} \)
41 \( 1 + (0.923 + 5.23i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-6.72 - 8.01i)T + (-7.46 + 42.3i)T^{2} \)
47 \( 1 + (-1.16 - 3.19i)T + (-36.0 + 30.2i)T^{2} \)
53 \( 1 + (-8.78 + 10.4i)T + (-9.20 - 52.1i)T^{2} \)
59 \( 1 + (9.41 + 3.42i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-6.94 - 5.83i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (3.73 + 10.2i)T + (-51.3 + 43.0i)T^{2} \)
71 \( 1 + (-0.519 + 0.435i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (-6.90 + 1.21i)T + (68.5 - 24.9i)T^{2} \)
79 \( 1 + (0.604 + 3.42i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-4.30 - 2.48i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.02 - 5.79i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (4.25 - 11.6i)T + (-74.3 - 62.3i)T^{2} \)
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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

−10.54199974876960623267071377331, −9.706129152254464900852388610773, −8.323946383894175098026462151280, −7.82231311187696200800725233500, −6.76097004105696532538263497060, −6.00825939648544170847636644732, −4.54831560202915831329510762874, −3.77637043176818229438245311735, −2.55067553201710106178233851328, −1.85621878158562782399995651358, 1.02750905276183180610500095839, 2.31212386084543424821148586175, 4.25685648230861890721659457562, 4.79017203761889466431368683841, 5.65965764720123507028413090240, 6.69707815226591437635645834236, 7.50180907350815971783225296672, 8.312998675864343805812185795080, 8.998952938223746125218693301558, 10.25361016953795635597721571942

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