Properties

Label 2-855-19.7-c1-0-18
Degree $2$
Conductor $855$
Sign $0.685 - 0.727i$
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  + (0.548 + 0.950i)2-s + (0.397 − 0.689i)4-s + (0.5 + 0.866i)5-s + 1.89·7-s + 3.06·8-s + (−0.548 + 0.950i)10-s − 0.134·11-s + (−1.75 + 3.04i)13-s + (1.03 + 1.79i)14-s + (0.887 + 1.53i)16-s + (−0.830 − 1.43i)17-s + (2.10 + 3.81i)19-s + 0.795·20-s + (−0.0737 − 0.127i)22-s + (2.68 − 4.65i)23-s + ⋯
L(s)  = 1  + (0.388 + 0.672i)2-s + (0.198 − 0.344i)4-s + (0.223 + 0.387i)5-s + 0.715·7-s + 1.08·8-s + (−0.173 + 0.300i)10-s − 0.0405·11-s + (−0.487 + 0.843i)13-s + (0.277 + 0.480i)14-s + (0.221 + 0.384i)16-s + (−0.201 − 0.348i)17-s + (0.483 + 0.875i)19-s + 0.177·20-s + (−0.0157 − 0.0272i)22-s + (0.559 − 0.969i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.685 - 0.727i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.685 - 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $0.685 - 0.727i$
Analytic conductor: \(6.82720\)
Root analytic conductor: \(2.61289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (406, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :1/2),\ 0.685 - 0.727i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.23219 + 0.963532i\)
\(L(\frac12)\) \(\approx\) \(2.23219 + 0.963532i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.10 - 3.81i)T \)
good2 \( 1 + (-0.548 - 0.950i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 - 1.89T + 7T^{2} \)
11 \( 1 + 0.134T + 11T^{2} \)
13 \( 1 + (1.75 - 3.04i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.830 + 1.43i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-2.68 + 4.65i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.48 + 4.30i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.56T + 31T^{2} \)
37 \( 1 + 1.69T + 37T^{2} \)
41 \( 1 + (-5.31 - 9.20i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.25 + 7.36i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.55 - 9.62i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.132 - 0.229i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.44 + 5.97i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.58 - 7.94i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.47 + 2.55i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.664 - 1.15i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.17 - 5.49i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.733 - 1.27i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 7.44T + 83T^{2} \)
89 \( 1 + (-4.86 + 8.43i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.73 + 15.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.22453604499348975760561031035, −9.604567332508242417436224099733, −8.347490649870989001068049045075, −7.57843959379373241300954341344, −6.69765455429910444246009069795, −6.07084536570485128008974747986, −4.95454937299164115662483613639, −4.37495236019104635016829500750, −2.69102888005893032253433713200, −1.48693864633134444710477465417, 1.30437448855195769159973321495, 2.52823987632570735448539823859, 3.49579383778879724002433990681, 4.74922015727192037354731648286, 5.24786527798256567969184610468, 6.69592229786919528951020586415, 7.64156745355855359002239226064, 8.290546563502448954922798898334, 9.307223203066508225825708953467, 10.32001733290957561811002224882

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