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Properties

Label 2-855-9.4-c1-0-57
Degree $2$
Conductor $855$
Sign $-0.00106 + 0.999i$
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  + (0.0983 + 0.170i)2-s + (0.994 − 1.41i)3-s + (0.980 − 1.69i)4-s + (0.5 − 0.866i)5-s + (0.339 + 0.0298i)6-s + (0.566 + 0.980i)7-s + 0.778·8-s + (−1.02 − 2.82i)9-s + 0.196·10-s + (0.420 + 0.728i)11-s + (−1.43 − 3.07i)12-s + (−0.596 + 1.03i)13-s + (−0.111 + 0.192i)14-s + (−0.731 − 1.57i)15-s + (−1.88 − 3.26i)16-s + 4.25·17-s + ⋯
L(s)  = 1  + (0.0695 + 0.120i)2-s + (0.574 − 0.818i)3-s + (0.490 − 0.849i)4-s + (0.223 − 0.387i)5-s + (0.138 + 0.0121i)6-s + (0.213 + 0.370i)7-s + 0.275·8-s + (−0.341 − 0.940i)9-s + 0.0621·10-s + (0.126 + 0.219i)11-s + (−0.413 − 0.889i)12-s + (−0.165 + 0.286i)13-s + (−0.0297 + 0.0515i)14-s + (−0.188 − 0.405i)15-s + (−0.471 − 0.816i)16-s + 1.03·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00106 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00106 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $-0.00106 + 0.999i$
Analytic conductor: \(6.82720\)
Root analytic conductor: \(2.61289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :1/2),\ -0.00106 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.64730 - 1.64905i\)
\(L(\frac12)\) \(\approx\) \(1.64730 - 1.64905i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.994 + 1.41i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 - T \)
good2 \( 1 + (-0.0983 - 0.170i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (-0.566 - 0.980i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.420 - 0.728i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.596 - 1.03i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 4.25T + 17T^{2} \)
23 \( 1 + (-3.94 + 6.83i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.14 - 1.98i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.69 - 8.12i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 8.54T + 37T^{2} \)
41 \( 1 + (2.95 - 5.11i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.975 - 1.69i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.25 + 3.90i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 3.93T + 53T^{2} \)
59 \( 1 + (-0.561 + 0.971i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.41 - 7.65i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.16 - 7.22i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.82T + 71T^{2} \)
73 \( 1 + 9.95T + 73T^{2} \)
79 \( 1 + (-0.113 - 0.196i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.07 - 13.9i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 13.8T + 89T^{2} \)
97 \( 1 + (0.938 + 1.62i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.911711272263049493825044959570, −8.999289134671891753126318862624, −8.363403988026709608749485759579, −7.17669096129767067104009933491, −6.68724297645797716560164127920, −5.62283382822401276428676335255, −4.86574486825839361984668183061, −3.21976005492316765620310119183, −2.06855960390736919897316947548, −1.12404714742509741161771187881, 1.96716923712017411884747484608, 3.23378675440457427536344627453, 3.65497657875539818900022440760, 4.92472376337513225122759296888, 5.95345714931357345899982297223, 7.40692227245883256358993810854, 7.65718354251554748346527724601, 8.751614318271310938232860323829, 9.583726471370372201091817093661, 10.42218574926420551126778073488

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