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Properties

Label 2-855-19.17-c1-0-23
Degree $2$
Conductor $855$
Sign $0.978 - 0.205i$
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  + (−0.370 + 2.09i)2-s + (−2.39 − 0.871i)4-s + (0.939 − 0.342i)5-s + (−0.742 − 1.28i)7-s + (0.583 − 1.01i)8-s + (0.370 + 2.09i)10-s + (2.34 − 4.05i)11-s + (−0.276 + 0.232i)13-s + (2.97 − 1.08i)14-s + (−1.99 − 1.67i)16-s + (0.951 − 5.39i)17-s + (−1.68 − 4.01i)19-s − 2.54·20-s + (7.64 + 6.41i)22-s + (−5.79 − 2.10i)23-s + ⋯
L(s)  = 1  + (−0.261 + 1.48i)2-s + (−1.19 − 0.435i)4-s + (0.420 − 0.152i)5-s + (−0.280 − 0.486i)7-s + (0.206 − 0.357i)8-s + (0.117 + 0.664i)10-s + (0.705 − 1.22i)11-s + (−0.0767 + 0.0643i)13-s + (0.795 − 0.289i)14-s + (−0.499 − 0.418i)16-s + (0.230 − 1.30i)17-s + (−0.386 − 0.922i)19-s − 0.569·20-s + (1.63 + 1.36i)22-s + (−1.20 − 0.439i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 - 0.205i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 - 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $0.978 - 0.205i$
Analytic conductor: \(6.82720\)
Root analytic conductor: \(2.61289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :1/2),\ 0.978 - 0.205i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11941 + 0.116085i\)
\(L(\frac12)\) \(\approx\) \(1.11941 + 0.116085i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.939 + 0.342i)T \)
19 \( 1 + (1.68 + 4.01i)T \)
good2 \( 1 + (0.370 - 2.09i)T + (-1.87 - 0.684i)T^{2} \)
7 \( 1 + (0.742 + 1.28i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.34 + 4.05i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.276 - 0.232i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-0.951 + 5.39i)T + (-15.9 - 5.81i)T^{2} \)
23 \( 1 + (5.79 + 2.10i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (-0.155 - 0.882i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (2.40 + 4.15i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 11.3T + 37T^{2} \)
41 \( 1 + (-4.01 - 3.36i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (6.78 - 2.46i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (-1.88 - 10.7i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + (6.12 + 2.23i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (-1.70 + 9.65i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (2.20 + 0.803i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-1.53 - 8.71i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (-6.02 + 2.19i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (-2.19 - 1.83i)T + (12.6 + 71.8i)T^{2} \)
79 \( 1 + (-1.58 - 1.32i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-3.08 - 5.33i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.54 + 2.13i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (-0.819 + 4.64i)T + (-91.1 - 33.1i)T^{2} \)
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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

−9.659460727450741212658006845319, −9.316815665882327028772739567093, −8.318649801986446788247024843215, −7.64129536548771001918308473107, −6.58707885770872725468195192087, −6.19126483847143022893788640949, −5.18896024685284704639033875444, −4.17899318653935803193009219180, −2.71385786529851258118248454990, −0.60758431616437159336866003649, 1.58541427038183005759710209637, 2.25831009564720472921758186900, 3.58355366788842276768368450137, 4.31010503496135081793176883396, 5.78462252899365405989098150863, 6.57623031329286742613188827760, 7.85613378288641392337528870714, 8.869259091434780322396083669129, 9.645693545209890672564012657874, 10.13993662303165631567460664323

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