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Properties

Label 2-855-19.7-c1-0-15
Degree $2$
Conductor $855$
Sign $0.488 - 0.872i$
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  + (0.690 + 1.19i)2-s + (0.0458 − 0.0794i)4-s + (0.5 + 0.866i)5-s − 4.36·7-s + 2.88·8-s + (−0.690 + 1.19i)10-s + 4.31·11-s + (3.18 − 5.51i)13-s + (−3.01 − 5.21i)14-s + (1.90 + 3.29i)16-s + (2.85 + 4.95i)17-s + (2.97 + 3.18i)19-s + 0.0917·20-s + (2.98 + 5.16i)22-s + (−0.289 + 0.501i)23-s + ⋯
L(s)  = 1  + (0.488 + 0.845i)2-s + (0.0229 − 0.0397i)4-s + (0.223 + 0.387i)5-s − 1.64·7-s + 1.02·8-s + (−0.218 + 0.378i)10-s + 1.30·11-s + (0.882 − 1.52i)13-s + (−0.805 − 1.39i)14-s + (0.476 + 0.824i)16-s + (0.693 + 1.20i)17-s + (0.681 + 0.731i)19-s + 0.0205·20-s + (0.635 + 1.10i)22-s + (−0.0604 + 0.104i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.488 - 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.488 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $0.488 - 0.872i$
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: \((2,\ 855,\ (\ :1/2),\ 0.488 - 0.872i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.92651 + 1.12966i\)
\(L(\frac12)\) \(\approx\) \(1.92651 + 1.12966i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.97 - 3.18i)T \)
good2 \( 1 + (-0.690 - 1.19i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + 4.36T + 7T^{2} \)
11 \( 1 - 4.31T + 11T^{2} \)
13 \( 1 + (-3.18 + 5.51i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.85 - 4.95i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (0.289 - 0.501i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.77 - 3.07i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.18T + 31T^{2} \)
37 \( 1 + 6.54T + 37T^{2} \)
41 \( 1 + (0.381 + 0.660i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.18 - 5.51i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.36 - 2.36i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.56 + 4.44i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.91 - 3.31i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.01 + 10.4i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.00 + 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.53 - 2.66i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (4.04 + 7.00i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.66 + 9.81i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 7.67T + 83T^{2} \)
89 \( 1 + (-4.92 + 8.52i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1 + 1.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.21904577147286472279683956479, −9.649463724282931846164675978484, −8.480472267083428402709164401723, −7.50012408391730537345725815714, −6.56760911047723808011929326156, −6.07551753297974820042337777982, −5.48351815867136827091332318335, −3.79204185665106746628438819864, −3.30045214767654285061351965676, −1.34601421350692043109777459324, 1.18293504847763992515418138062, 2.58929031955225532426223690092, 3.62109176139026843101015426110, 4.20735372439016246549310961310, 5.55174736425381897018626541603, 6.74456207727086610366391668272, 7.08024606984096454570633404654, 8.702972254313098748613615542308, 9.422615702062892633012434758962, 9.906153919586727420529262803742

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