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Properties

Label 2-855-285.284-c1-0-25
Degree $2$
Conductor $855$
Sign $0.747 - 0.663i$
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  + 1.94i·2-s − 1.78·4-s + (1.33 − 1.79i)5-s − 1.78i·7-s + 0.410i·8-s + (3.48 + 2.60i)10-s − 2.08i·11-s − 0.735·13-s + 3.47·14-s − 4.37·16-s + 3.58·17-s + (−1.74 − 3.99i)19-s + (−2.39 + 3.20i)20-s + 4.06·22-s + 8.39·23-s + ⋯
L(s)  = 1  + 1.37i·2-s − 0.894·4-s + (0.598 − 0.801i)5-s − 0.674i·7-s + 0.145i·8-s + (1.10 + 0.823i)10-s − 0.630i·11-s − 0.203·13-s + 0.928·14-s − 1.09·16-s + 0.868·17-s + (−0.400 − 0.916i)19-s + (−0.535 + 0.716i)20-s + 0.867·22-s + 1.75·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.747 - 0.663i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.747 - 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $0.747 - 0.663i$
Analytic conductor: \(6.82720\)
Root analytic conductor: \(2.61289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (854, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :1/2),\ 0.747 - 0.663i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.63600 + 0.621303i\)
\(L(\frac12)\) \(\approx\) \(1.63600 + 0.621303i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.33 + 1.79i)T \)
19 \( 1 + (1.74 + 3.99i)T \)
good2 \( 1 - 1.94iT - 2T^{2} \)
7 \( 1 + 1.78iT - 7T^{2} \)
11 \( 1 + 2.08iT - 11T^{2} \)
13 \( 1 + 0.735T + 13T^{2} \)
17 \( 1 - 3.58T + 17T^{2} \)
23 \( 1 - 8.39T + 23T^{2} \)
29 \( 1 - 6.63T + 29T^{2} \)
31 \( 1 - 1.76iT - 31T^{2} \)
37 \( 1 - 6.82T + 37T^{2} \)
41 \( 1 + 2.17T + 41T^{2} \)
43 \( 1 - 1.11iT - 43T^{2} \)
47 \( 1 + 1.05T + 47T^{2} \)
53 \( 1 + 1.53iT - 53T^{2} \)
59 \( 1 - 13.0T + 59T^{2} \)
61 \( 1 + 11.3T + 61T^{2} \)
67 \( 1 + 9.41T + 67T^{2} \)
71 \( 1 + 15.5T + 71T^{2} \)
73 \( 1 - 7.96iT - 73T^{2} \)
79 \( 1 + 4.91iT - 79T^{2} \)
83 \( 1 - 2.63T + 83T^{2} \)
89 \( 1 + 8.34T + 89T^{2} \)
97 \( 1 - 8.41T + 97T^{2} \)
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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

−10.08560473403957229118183158271, −9.039454078196468413009022557410, −8.555284210827511120269443649403, −7.60384367651166252755288402346, −6.82407655702245221899672608987, −6.00149403096395315612121895470, −5.13142343931574597354153002939, −4.46253008427053330897252470697, −2.81523067220699504664480318887, −0.970341108690200332307886059714, 1.43760520797564287866374798469, 2.53883464946426989492935061557, 3.17374463216151733463918100472, 4.44319743168148661046953396850, 5.60647853452664205778741367373, 6.58997294338143970323200975367, 7.49984082740201039203610529503, 8.792194772540644980751644984291, 9.605440131819404511372547113490, 10.21121441527562795408917300291

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