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Properties

Label 2-855-19.11-c1-0-14
Degree $2$
Conductor $855$
Sign $0.999 - 0.0417i$
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  + (0.0850 − 0.147i)2-s + (0.985 + 1.70i)4-s + (0.5 − 0.866i)5-s − 0.218·7-s + 0.675·8-s + (−0.0850 − 0.147i)10-s + 5.33·11-s + (−3.01 − 5.22i)13-s + (−0.0185 + 0.0321i)14-s + (−1.91 + 3.31i)16-s + (3.27 − 5.67i)17-s + (1.50 + 4.09i)19-s + 1.97·20-s + (0.453 − 0.785i)22-s + (1.90 + 3.30i)23-s + ⋯
L(s)  = 1  + (0.0601 − 0.104i)2-s + (0.492 + 0.853i)4-s + (0.223 − 0.387i)5-s − 0.0825·7-s + 0.238·8-s + (−0.0268 − 0.0465i)10-s + 1.60·11-s + (−0.836 − 1.44i)13-s + (−0.00496 + 0.00860i)14-s + (−0.478 + 0.828i)16-s + (0.794 − 1.37i)17-s + (0.345 + 0.938i)19-s + 0.440·20-s + (0.0966 − 0.167i)22-s + (0.397 + 0.689i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0417i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0417i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $0.999 - 0.0417i$
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: \((2,\ 855,\ (\ :1/2),\ 0.999 - 0.0417i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.03289 + 0.0424918i\)
\(L(\frac12)\) \(\approx\) \(2.03289 + 0.0424918i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.50 - 4.09i)T \)
good2 \( 1 + (-0.0850 + 0.147i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 + 0.218T + 7T^{2} \)
11 \( 1 - 5.33T + 11T^{2} \)
13 \( 1 + (3.01 + 5.22i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.27 + 5.67i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-1.90 - 3.30i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.525 - 0.909i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.65T + 31T^{2} \)
37 \( 1 - 3.10T + 37T^{2} \)
41 \( 1 + (1.75 - 3.03i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.39 - 2.41i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.26 - 7.38i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.57 + 4.45i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.95 + 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.25 - 3.91i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.30 - 5.71i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.12 + 5.40i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.31 - 5.74i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.80 - 3.13i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 8.39T + 83T^{2} \)
89 \( 1 + (4.79 + 8.29i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.95 + 13.7i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.953289606019304089500838217812, −9.510903217294738180585519145374, −8.364951721002578984497243410103, −7.64836342076534396298954561221, −6.88053966751201850294300026186, −5.81524880731638450660110029078, −4.80323362849744691577159634161, −3.58730841260406934012360040675, −2.81418465857582803367015090293, −1.24975165779074735322755676061, 1.32680363263492267668899289761, 2.38760356833666954384145755479, 3.87416635042810286556632926427, 4.86330314736239404175865494988, 6.05469552692129847617623681864, 6.66966530748986598225284701723, 7.23079793001023838767644092678, 8.686521683685868549360832291761, 9.482340025346449957367918822498, 10.08163960406639897753519748288

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