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Properties

Label 2-855-95.44-c1-0-4
Degree $2$
Conductor $855$
Sign $0.718 + 0.695i$
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  + (−1.14 − 1.36i)2-s + (−0.205 + 1.16i)4-s + (−1.67 + 1.47i)5-s + (−3.67 − 2.11i)7-s + (−1.25 + 0.727i)8-s + (3.94 + 0.596i)10-s + (−0.245 − 0.425i)11-s + (1.42 + 3.91i)13-s + (1.31 + 7.45i)14-s + (4.66 + 1.69i)16-s + (1.30 + 1.55i)17-s + (−1.86 − 3.93i)19-s + (−1.38 − 2.26i)20-s + (−0.299 + 0.823i)22-s + (4.32 + 0.763i)23-s + ⋯
L(s)  = 1  + (−0.811 − 0.966i)2-s + (−0.102 + 0.583i)4-s + (−0.749 + 0.661i)5-s + (−1.38 − 0.801i)7-s + (−0.445 + 0.257i)8-s + (1.24 + 0.188i)10-s + (−0.0740 − 0.128i)11-s + (0.394 + 1.08i)13-s + (0.351 + 1.99i)14-s + (1.16 + 0.424i)16-s + (0.317 + 0.378i)17-s + (−0.428 − 0.903i)19-s + (−0.308 − 0.505i)20-s + (−0.0638 + 0.175i)22-s + (0.902 + 0.159i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.718 + 0.695i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.718 + 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $0.718 + 0.695i$
Analytic conductor: \(6.82720\)
Root analytic conductor: \(2.61289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (424, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :1/2),\ 0.718 + 0.695i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.531321 - 0.215102i\)
\(L(\frac12)\) \(\approx\) \(0.531321 - 0.215102i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.67 - 1.47i)T \)
19 \( 1 + (1.86 + 3.93i)T \)
good2 \( 1 + (1.14 + 1.36i)T + (-0.347 + 1.96i)T^{2} \)
7 \( 1 + (3.67 + 2.11i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.245 + 0.425i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.42 - 3.91i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (-1.30 - 1.55i)T + (-2.95 + 16.7i)T^{2} \)
23 \( 1 + (-4.32 - 0.763i)T + (21.6 + 7.86i)T^{2} \)
29 \( 1 + (2.49 + 2.09i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (2.04 - 3.54i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.14iT - 37T^{2} \)
41 \( 1 + (-4.10 - 1.49i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-10.4 + 1.84i)T + (40.4 - 14.7i)T^{2} \)
47 \( 1 + (-1.68 + 2.00i)T + (-8.16 - 46.2i)T^{2} \)
53 \( 1 + (-11.2 - 1.98i)T + (49.8 + 18.1i)T^{2} \)
59 \( 1 + (0.415 - 0.348i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (2.36 - 13.4i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-4.60 + 5.48i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (1.04 + 5.94i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (0.702 - 1.93i)T + (-55.9 - 46.9i)T^{2} \)
79 \( 1 + (-5.01 - 1.82i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (7.02 + 4.05i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.07 + 1.48i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (-3.62 - 4.32i)T + (-16.8 + 95.5i)T^{2} \)
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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

−10.27823371757033662770593945830, −9.280883794842891939330797089358, −8.814092755228246897926395245512, −7.49890867557039858596367519657, −6.83356249217638995785902832039, −5.94733213772338678796847009717, −4.16631436593673985167368378376, −3.41947870942050037754955493037, −2.47041784901754950823703858366, −0.74810556718707225720616954909, 0.59603797698377910004505947187, 2.92526926510872376699210788435, 3.81694954039279945326347225392, 5.43327079047102938286151653409, 6.00292259705265381493846450215, 7.04367923963604530364845102924, 7.80494524964872235450843645119, 8.558697062878484188883946390887, 9.219708443551286445848123658011, 9.848183049567461505594368725617

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