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Properties

Label 2-855-19.11-c1-0-27
Degree $2$
Conductor $855$
Sign $-0.868 + 0.495i$
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  + (1.12 − 1.94i)2-s + (−1.52 − 2.64i)4-s + (0.5 − 0.866i)5-s + 3.16·7-s − 2.36·8-s + (−1.12 − 1.94i)10-s − 4.81·11-s + (−0.583 − 1.01i)13-s + (3.55 − 6.16i)14-s + (0.396 − 0.686i)16-s + (2.92 − 5.06i)17-s + (3.21 + 2.94i)19-s − 3.05·20-s + (−5.41 + 9.37i)22-s + (−4.29 − 7.44i)23-s + ⋯
L(s)  = 1  + (0.794 − 1.37i)2-s + (−0.762 − 1.32i)4-s + (0.223 − 0.387i)5-s + 1.19·7-s − 0.835·8-s + (−0.355 − 0.615i)10-s − 1.45·11-s + (−0.161 − 0.280i)13-s + (0.950 − 1.64i)14-s + (0.0990 − 0.171i)16-s + (0.708 − 1.22i)17-s + (0.737 + 0.675i)19-s − 0.682·20-s + (−1.15 + 1.99i)22-s + (−0.896 − 1.55i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.868 + 0.495i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.868 + 0.495i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $-0.868 + 0.495i$
Analytic conductor: \(6.82720\)
Root analytic conductor: \(2.61289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (676, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :1/2),\ -0.868 + 0.495i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.647549 - 2.43957i\)
\(L(\frac12)\) \(\approx\) \(0.647549 - 2.43957i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-3.21 - 2.94i)T \)
good2 \( 1 + (-1.12 + 1.94i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 - 3.16T + 7T^{2} \)
11 \( 1 + 4.81T + 11T^{2} \)
13 \( 1 + (0.583 + 1.01i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.92 + 5.06i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (4.29 + 7.44i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.65 - 6.33i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.10T + 31T^{2} \)
37 \( 1 - 7.26T + 37T^{2} \)
41 \( 1 + (1.24 - 2.16i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.583 - 1.01i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.68 - 8.12i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.37 + 2.38i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.05 - 8.75i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.55 - 4.42i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.519 + 0.899i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.16 - 3.74i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.81 - 3.14i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.53 + 13.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 8.98T + 83T^{2} \)
89 \( 1 + (-2.08 - 3.61i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1 - 1.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.27322471871210881657830907846, −9.300715292479435894385509091941, −8.081753278215250545635510208017, −7.54320999680675741094129115277, −5.74389410563775845858683484203, −5.03513142938747982631557615935, −4.50845373711294699659980631196, −3.09299939021671678124670984557, −2.27251685133949532091314124951, −1.02239810188298763216163750387, 1.97884053621999015866306580090, 3.52062454948250462706987725656, 4.61985666110948347853193152424, 5.43516119791464786853015787362, 5.94426295325080117401558662472, 7.18240358422440433888852134393, 7.86016974637340344133961469180, 8.213681131149268730231029648237, 9.625032063389675403390709252059, 10.56137831720619650942021828242

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