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Properties

Label 2-855-19.7-c1-0-6
Degree $2$
Conductor $855$
Sign $0.997 + 0.0653i$
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  + (−1.33 − 2.30i)2-s + (−2.54 + 4.41i)4-s + (−0.5 − 0.866i)5-s + 4.51·7-s + 8.25·8-s + (−1.33 + 2.30i)10-s − 1.88·11-s + (−2.21 + 3.83i)13-s + (−6.01 − 10.4i)14-s + (−5.90 − 10.2i)16-s + (0.818 + 1.41i)17-s + (−1.60 + 4.05i)19-s + 5.09·20-s + (2.50 + 4.33i)22-s + (−3.47 + 6.01i)23-s + ⋯
L(s)  = 1  + (−0.942 − 1.63i)2-s + (−1.27 + 2.20i)4-s + (−0.223 − 0.387i)5-s + 1.70·7-s + 2.91·8-s + (−0.421 + 0.729i)10-s − 0.566·11-s + (−0.614 + 1.06i)13-s + (−1.60 − 2.78i)14-s + (−1.47 − 2.55i)16-s + (0.198 + 0.343i)17-s + (−0.367 + 0.930i)19-s + 1.14·20-s + (0.534 + 0.925i)22-s + (−0.724 + 1.25i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0653i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0653i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $0.997 + 0.0653i$
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: \((2,\ 855,\ (\ :1/2),\ 0.997 + 0.0653i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.673343 - 0.0220313i\)
\(L(\frac12)\) \(\approx\) \(0.673343 - 0.0220313i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.60 - 4.05i)T \)
good2 \( 1 + (1.33 + 2.30i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 - 4.51T + 7T^{2} \)
11 \( 1 + 1.88T + 11T^{2} \)
13 \( 1 + (2.21 - 3.83i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.818 - 1.41i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (3.47 - 6.01i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.37 - 5.84i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 2.32T + 31T^{2} \)
37 \( 1 + 7.21T + 37T^{2} \)
41 \( 1 + (-3.12 - 5.41i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.41 - 2.45i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.67 + 8.09i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.90 - 10.2i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.24 + 2.15i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.28 + 12.6i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.479 - 0.830i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.80 - 4.84i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.664 + 1.15i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.46 - 2.53i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 12.9T + 83T^{2} \)
89 \( 1 + (-2.43 + 4.21i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-9.04 - 15.6i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.29786771302250783039627013596, −9.419528248333005941415486966066, −8.647378805757972903154113045635, −7.959010969456390585428671657021, −7.40013686909979628046344728911, −5.34983717476667989858940579111, −4.45489426669779897020671537700, −3.60265810749828375268073536457, −2.03354211404618427577945470666, −1.51319904943922016864213002358, 0.46606897296157606137777679760, 2.23452174727910705146739364408, 4.42345554238175512835872910439, 5.16750077731802944260361442990, 5.89847923966391053986945179339, 7.13955480053536291475382163836, 7.65259352086464607719497454659, 8.258979163704972882831102674804, 8.937623257684669584530068180883, 10.12723924889647813219635214058

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