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Properties

Label 2-855-285.284-c1-0-22
Degree $2$
Conductor $855$
Sign $0.241 - 0.970i$
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  + 2.62i·2-s − 4.86·4-s + (2.20 + 0.365i)5-s − 3.50i·7-s − 7.51i·8-s + (−0.958 + 5.78i)10-s − 3.68i·11-s − 2.74·13-s + 9.19·14-s + 9.94·16-s + 3.56·17-s + (3.34 + 2.79i)19-s + (−10.7 − 1.78i)20-s + 9.66·22-s − 1.83·23-s + ⋯
L(s)  = 1  + 1.85i·2-s − 2.43·4-s + (0.986 + 0.163i)5-s − 1.32i·7-s − 2.65i·8-s + (−0.303 + 1.82i)10-s − 1.11i·11-s − 0.761·13-s + 2.45·14-s + 2.48·16-s + 0.864·17-s + (0.768 + 0.640i)19-s + (−2.40 − 0.398i)20-s + 2.06·22-s − 0.382·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.241 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.241 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $0.241 - 0.970i$
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: \((2,\ 855,\ (\ :1/2),\ 0.241 - 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20874 + 0.945221i\)
\(L(\frac12)\) \(\approx\) \(1.20874 + 0.945221i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.20 - 0.365i)T \)
19 \( 1 + (-3.34 - 2.79i)T \)
good2 \( 1 - 2.62iT - 2T^{2} \)
7 \( 1 + 3.50iT - 7T^{2} \)
11 \( 1 + 3.68iT - 11T^{2} \)
13 \( 1 + 2.74T + 13T^{2} \)
17 \( 1 - 3.56T + 17T^{2} \)
23 \( 1 + 1.83T + 23T^{2} \)
29 \( 1 - 10.0T + 29T^{2} \)
31 \( 1 + 2.21iT - 31T^{2} \)
37 \( 1 - 5.95T + 37T^{2} \)
41 \( 1 - 5.24T + 41T^{2} \)
43 \( 1 + 2.00iT - 43T^{2} \)
47 \( 1 + 8.52T + 47T^{2} \)
53 \( 1 + 10.1iT - 53T^{2} \)
59 \( 1 + 12.4T + 59T^{2} \)
61 \( 1 + 0.203T + 61T^{2} \)
67 \( 1 + 10.1T + 67T^{2} \)
71 \( 1 - 9.26T + 71T^{2} \)
73 \( 1 + 3.53iT - 73T^{2} \)
79 \( 1 - 13.0iT - 79T^{2} \)
83 \( 1 - 11.3T + 83T^{2} \)
89 \( 1 - 10.2T + 89T^{2} \)
97 \( 1 + 18.9T + 97T^{2} \)
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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

−9.917651879420715668034979765427, −9.515256753160891526021443646892, −8.258070428488455253689795293299, −7.74126275066825643350954561746, −6.83171014924259432813506080270, −6.14597152358257520538502695603, −5.38132598509370471714335476614, −4.46230251272627149306521090011, −3.26021034629164188686815153104, −0.885428573623812458777254141788, 1.34532724975673793982671563369, 2.43619978244141838590284695015, 2.93084071471178247919963936352, 4.63853349684701527940484091258, 5.11320712726229272409205631705, 6.21670671453493016379658009855, 7.78493120486535109340990492460, 8.917358611245662777205715401935, 9.496043453040633766531192967857, 9.937001891924220230938816427979

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