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Properties

Label 2-855-95.24-c1-0-1
Degree $2$
Conductor $855$
Sign $0.306 + 0.951i$
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  + (0.810 + 2.22i)2-s + (−2.77 + 2.32i)4-s + (−2.12 + 0.703i)5-s + (−1.41 + 0.818i)7-s + (−3.32 − 1.92i)8-s + (−3.28 − 4.15i)10-s + (1.36 − 2.36i)11-s + (−6.10 + 1.07i)13-s + (−2.97 − 2.49i)14-s + (0.325 − 1.84i)16-s + (1.06 + 2.93i)17-s + (3.46 − 2.64i)19-s + (4.25 − 6.89i)20-s + (6.36 + 1.12i)22-s + (−4.69 − 5.59i)23-s + ⋯
L(s)  = 1  + (0.573 + 1.57i)2-s + (−1.38 + 1.16i)4-s + (−0.949 + 0.314i)5-s + (−0.535 + 0.309i)7-s + (−1.17 − 0.679i)8-s + (−1.03 − 1.31i)10-s + (0.411 − 0.712i)11-s + (−1.69 + 0.298i)13-s + (−0.794 − 0.666i)14-s + (0.0812 − 0.460i)16-s + (0.259 + 0.712i)17-s + (0.795 − 0.605i)19-s + (0.950 − 1.54i)20-s + (1.35 + 0.239i)22-s + (−0.978 − 1.16i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.306 + 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.306 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $0.306 + 0.951i$
Analytic conductor: \(6.82720\)
Root analytic conductor: \(2.61289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (784, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :1/2),\ 0.306 + 0.951i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.280272 - 0.204224i\)
\(L(\frac12)\) \(\approx\) \(0.280272 - 0.204224i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.12 - 0.703i)T \)
19 \( 1 + (-3.46 + 2.64i)T \)
good2 \( 1 + (-0.810 - 2.22i)T + (-1.53 + 1.28i)T^{2} \)
7 \( 1 + (1.41 - 0.818i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.36 + 2.36i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (6.10 - 1.07i)T + (12.2 - 4.44i)T^{2} \)
17 \( 1 + (-1.06 - 2.93i)T + (-13.0 + 10.9i)T^{2} \)
23 \( 1 + (4.69 + 5.59i)T + (-3.99 + 22.6i)T^{2} \)
29 \( 1 + (2.09 + 0.761i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (2.21 + 3.83i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 2.04iT - 37T^{2} \)
41 \( 1 + (0.681 - 3.86i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (0.303 - 0.362i)T + (-7.46 - 42.3i)T^{2} \)
47 \( 1 + (-0.787 + 2.16i)T + (-36.0 - 30.2i)T^{2} \)
53 \( 1 + (-4.09 - 4.87i)T + (-9.20 + 52.1i)T^{2} \)
59 \( 1 + (11.5 - 4.19i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (3.66 - 3.07i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-0.229 + 0.630i)T + (-51.3 - 43.0i)T^{2} \)
71 \( 1 + (2.01 + 1.69i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (6.41 + 1.13i)T + (68.5 + 24.9i)T^{2} \)
79 \( 1 + (0.715 - 4.05i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (5.56 - 3.21i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-3.00 - 17.0i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (0.0444 + 0.122i)T + (-74.3 + 62.3i)T^{2} \)
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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

−10.89163716986331102917777747416, −9.732800464041281230677461191575, −8.839199130258332373611285926198, −7.956999844709876786299194558091, −7.36628737709323829970299444511, −6.55644988828274358889036469483, −5.82737879972879977189398761528, −4.73030038926788496701919851466, −3.97344981700779969915318510347, −2.82632951403890989485656344485, 0.13664241599455164961653086752, 1.70178012678579208139555433936, 3.05100090199233759666332855011, 3.76673570522763379344121195330, 4.69874374330333806864434052847, 5.43304680634011266829409725322, 7.18910722897413053200523024442, 7.64489584612806323316497524617, 9.177609217335886705298583550633, 9.775860744299691771622603031804

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