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Properties

Label 2-855-19.11-c1-0-31
Degree $2$
Conductor $855$
Sign $-0.370 - 0.928i$
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  + (1.37 − 2.38i)2-s + (−2.80 − 4.85i)4-s + (0.5 − 0.866i)5-s − 2.84·7-s − 9.94·8-s + (−1.37 − 2.38i)10-s + 0.864·11-s + (−0.321 − 0.557i)13-s + (−3.92 + 6.80i)14-s + (−8.11 + 14.0i)16-s + (1.87 − 3.24i)17-s + (−3.36 + 2.77i)19-s − 5.60·20-s + (1.19 − 2.06i)22-s + (0.208 + 0.361i)23-s + ⋯
L(s)  = 1  + (0.975 − 1.68i)2-s + (−1.40 − 2.42i)4-s + (0.223 − 0.387i)5-s − 1.07·7-s − 3.51·8-s + (−0.436 − 0.755i)10-s + 0.260·11-s + (−0.0892 − 0.154i)13-s + (−1.04 + 1.81i)14-s + (−2.02 + 3.51i)16-s + (0.453 − 0.785i)17-s + (−0.770 + 0.636i)19-s − 1.25·20-s + (0.254 − 0.440i)22-s + (0.0435 + 0.0753i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.370 - 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.370 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $-0.370 - 0.928i$
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: \((2,\ 855,\ (\ :1/2),\ -0.370 - 0.928i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.835259 + 1.23256i\)
\(L(\frac12)\) \(\approx\) \(0.835259 + 1.23256i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.36 - 2.77i)T \)
good2 \( 1 + (-1.37 + 2.38i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 + 2.84T + 7T^{2} \)
11 \( 1 - 0.864T + 11T^{2} \)
13 \( 1 + (0.321 + 0.557i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.87 + 3.24i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-0.208 - 0.361i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.85 + 8.40i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.93T + 31T^{2} \)
37 \( 1 - 6.36T + 37T^{2} \)
41 \( 1 + (2.00 - 3.47i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.02 + 1.78i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.97 + 3.42i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.49 + 9.51i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.22 + 2.13i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.16 + 5.48i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.26 - 2.19i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.891 - 1.54i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3.56 + 6.17i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.912 - 1.58i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 7.43T + 83T^{2} \)
89 \( 1 + (-2.22 - 3.85i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.42 + 9.39i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.605534692094072041527012983893, −9.544169734691492201627223314167, −8.196715362965055267513701145512, −6.48272029808297644737798805935, −5.82365582275884945487578511500, −4.81963481344565008355690764325, −3.90083547441996642315630084675, −3.03879938354027395747823524729, −1.99218221040240746535129833470, −0.51066131467419042088468104233, 2.87918855079306428133464893516, 3.73920699839670109988785380090, 4.69823984405448521196899132880, 5.81842266812332649424729689534, 6.39656746448314994907894469208, 7.01838934902261350171524738534, 7.896955081016784288208372947880, 8.886583705809772140148935348959, 9.515365227349482141921561761963, 10.72505704250787658259090750828

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