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Properties

Label 2-855-95.4-c1-0-11
Degree $2$
Conductor $855$
Sign $-0.988 + 0.153i$
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  + (−0.810 + 2.22i)2-s + (−2.77 − 2.32i)4-s + (0.323 + 2.21i)5-s + (1.41 + 0.818i)7-s + (3.32 − 1.92i)8-s + (−5.19 − 1.07i)10-s + (1.36 + 2.36i)11-s + (6.10 + 1.07i)13-s + (−2.97 + 2.49i)14-s + (0.325 + 1.84i)16-s + (−1.06 + 2.93i)17-s + (3.46 + 2.64i)19-s + (4.25 − 6.89i)20-s + (−6.36 + 1.12i)22-s + (4.69 − 5.59i)23-s + ⋯
L(s)  = 1  + (−0.573 + 1.57i)2-s + (−1.38 − 1.16i)4-s + (0.144 + 0.989i)5-s + (0.535 + 0.309i)7-s + (1.17 − 0.679i)8-s + (−1.64 − 0.339i)10-s + (0.411 + 0.712i)11-s + (1.69 + 0.298i)13-s + (−0.794 + 0.666i)14-s + (0.0812 + 0.460i)16-s + (−0.259 + 0.712i)17-s + (0.795 + 0.605i)19-s + (0.950 − 1.54i)20-s + (−1.35 + 0.239i)22-s + (0.978 − 1.16i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.153i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.988 + 0.153i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $-0.988 + 0.153i$
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: \((2,\ 855,\ (\ :1/2),\ -0.988 + 0.153i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0938421 - 1.21792i\)
\(L(\frac12)\) \(\approx\) \(0.0938421 - 1.21792i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.323 - 2.21i)T \)
19 \( 1 + (-3.46 - 2.64i)T \)
good2 \( 1 + (0.810 - 2.22i)T + (-1.53 - 1.28i)T^{2} \)
7 \( 1 + (-1.41 - 0.818i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.36 - 2.36i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-6.10 - 1.07i)T + (12.2 + 4.44i)T^{2} \)
17 \( 1 + (1.06 - 2.93i)T + (-13.0 - 10.9i)T^{2} \)
23 \( 1 + (-4.69 + 5.59i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (2.09 - 0.761i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (2.21 - 3.83i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2.04iT - 37T^{2} \)
41 \( 1 + (0.681 + 3.86i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-0.303 - 0.362i)T + (-7.46 + 42.3i)T^{2} \)
47 \( 1 + (0.787 + 2.16i)T + (-36.0 + 30.2i)T^{2} \)
53 \( 1 + (4.09 - 4.87i)T + (-9.20 - 52.1i)T^{2} \)
59 \( 1 + (11.5 + 4.19i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (3.66 + 3.07i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (0.229 + 0.630i)T + (-51.3 + 43.0i)T^{2} \)
71 \( 1 + (2.01 - 1.69i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (-6.41 + 1.13i)T + (68.5 - 24.9i)T^{2} \)
79 \( 1 + (0.715 + 4.05i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-5.56 - 3.21i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-3.00 + 17.0i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-0.0444 + 0.122i)T + (-74.3 - 62.3i)T^{2} \)
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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

−10.49317739189935433587482933066, −9.436987990033297910364897102493, −8.734113661613094776577980331651, −8.012189228699797795341718249824, −7.10496475666104642065240944045, −6.45099999655361612554450279275, −5.81409382408294252246081368941, −4.70181734517742122877099244138, −3.43119197584761775537348664078, −1.63732340099564892946428297362, 0.847158744661382320170656048795, 1.53188001638765260387852677470, 3.09966831573889786094271330858, 3.93405602862424143905022736268, 4.99137206177378845472587665974, 6.06341664214531876277722534611, 7.62806533522322164087203663736, 8.457086457176684721023611109124, 9.186438596292034596010364179595, 9.559477595418506599140452419365

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