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Properties

Label 2-855-95.64-c1-0-30
Degree $2$
Conductor $855$
Sign $0.263 + 0.964i$
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  + (−0.408 + 0.235i)2-s + (−0.888 + 1.53i)4-s + (−1.47 + 1.67i)5-s − 1.17i·7-s − 1.78i·8-s + (0.208 − 1.03i)10-s − 0.713·11-s + (−3.55 − 2.05i)13-s + (0.277 + 0.480i)14-s + (−1.35 − 2.34i)16-s + (2.21 − 1.27i)17-s + (−1.57 + 4.06i)19-s + (−1.26 − 3.76i)20-s + (0.291 − 0.168i)22-s + (−0.525 − 0.303i)23-s + ⋯
L(s)  = 1  + (−0.288 + 0.166i)2-s + (−0.444 + 0.769i)4-s + (−0.661 + 0.750i)5-s − 0.444i·7-s − 0.630i·8-s + (0.0659 − 0.327i)10-s − 0.215·11-s + (−0.985 − 0.569i)13-s + (0.0741 + 0.128i)14-s + (−0.339 − 0.587i)16-s + (0.536 − 0.309i)17-s + (−0.360 + 0.932i)19-s + (−0.283 − 0.842i)20-s + (0.0621 − 0.0358i)22-s + (−0.109 − 0.0632i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.263 + 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.263 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $0.263 + 0.964i$
Analytic conductor: \(6.82720\)
Root analytic conductor: \(2.61289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :1/2),\ 0.263 + 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.337637 - 0.257688i\)
\(L(\frac12)\) \(\approx\) \(0.337637 - 0.257688i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.47 - 1.67i)T \)
19 \( 1 + (1.57 - 4.06i)T \)
good2 \( 1 + (0.408 - 0.235i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + 1.17iT - 7T^{2} \)
11 \( 1 + 0.713T + 11T^{2} \)
13 \( 1 + (3.55 + 2.05i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.21 + 1.27i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (0.525 + 0.303i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.429 + 0.744i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.50T + 31T^{2} \)
37 \( 1 + 9.38iT - 37T^{2} \)
41 \( 1 + (2.06 + 3.57i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-8.76 + 5.06i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (9.10 + 5.25i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.31 - 2.49i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.12 - 5.41i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.27 + 3.94i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.48 + 2.59i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.58 + 11.4i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (10.7 - 6.18i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.98 + 5.16i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 13.8iT - 83T^{2} \)
89 \( 1 + (7.98 - 13.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (14.4 - 8.35i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.15019659084206292820716081571, −9.077312346733545291177694890327, −8.074527346335580756776805917748, −7.53592246274547054528945304202, −6.95857852599919749537712363881, −5.63381778867113160796921255080, −4.36616721329291164770217427992, −3.60624581296481631699356376772, −2.59705722490137885243240189646, −0.25019986941075926443052629128, 1.27548412626321948262623124096, 2.69315131063606193054837544945, 4.33648998527922476530527725810, 4.91123830460233457975224293879, 5.82883482110503582042313162010, 6.98454440228631800043252829077, 8.097783903865524288932697667165, 8.719522859597138084719958849236, 9.528887482771539141803224327716, 10.13364747244004576098555499924

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