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Properties

Label 2-855-5.4-c1-0-21
Degree $2$
Conductor $855$
Sign $0.990 - 0.139i$
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  − 0.311i·2-s + 1.90·4-s + (2.21 − 0.311i)5-s + 2.68i·7-s − 1.21i·8-s + (−0.0967 − 0.688i)10-s + 0.214·11-s + 3.11i·13-s + 0.836·14-s + 3.42·16-s − 0.474i·17-s + 19-s + (4.21 − 0.592i)20-s − 0.0666i·22-s + 2.47i·23-s + ⋯
L(s)  = 1  − 0.219i·2-s + 0.951·4-s + (0.990 − 0.139i)5-s + 1.01i·7-s − 0.429i·8-s + (−0.0306 − 0.217i)10-s + 0.0646·11-s + 0.864i·13-s + 0.223·14-s + 0.857·16-s − 0.115i·17-s + 0.229·19-s + (0.942 − 0.132i)20-s − 0.0142i·22-s + 0.515i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.139i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 - 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $0.990 - 0.139i$
Analytic conductor: \(6.82720\)
Root analytic conductor: \(2.61289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (514, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :1/2),\ 0.990 - 0.139i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.32899 + 0.162810i\)
\(L(\frac12)\) \(\approx\) \(2.32899 + 0.162810i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.21 + 0.311i)T \)
19 \( 1 - T \)
good2 \( 1 + 0.311iT - 2T^{2} \)
7 \( 1 - 2.68iT - 7T^{2} \)
11 \( 1 - 0.214T + 11T^{2} \)
13 \( 1 - 3.11iT - 13T^{2} \)
17 \( 1 + 0.474iT - 17T^{2} \)
23 \( 1 - 2.47iT - 23T^{2} \)
29 \( 1 + 2.83T + 29T^{2} \)
31 \( 1 + 4.90T + 31T^{2} \)
37 \( 1 - 1.93iT - 37T^{2} \)
41 \( 1 + 5.45T + 41T^{2} \)
43 \( 1 + 10.7iT - 43T^{2} \)
47 \( 1 + 9.52iT - 47T^{2} \)
53 \( 1 - 13.7iT - 53T^{2} \)
59 \( 1 + 6.56T + 59T^{2} \)
61 \( 1 - 7.76T + 61T^{2} \)
67 \( 1 + 6.85iT - 67T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + 16.6iT - 73T^{2} \)
79 \( 1 + 1.37T + 79T^{2} \)
83 \( 1 - 1.65iT - 83T^{2} \)
89 \( 1 - 16.5T + 89T^{2} \)
97 \( 1 - 9.05iT - 97T^{2} \)
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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

−10.22845416909005957501234482016, −9.325874843440256882364172395553, −8.800441599846267655711984166740, −7.49492536072777670710115303382, −6.64371180605065553773249462306, −5.85497684371161384596027040632, −5.13393840319861952773988949093, −3.54721541588602838117076406496, −2.36087723735763536132629394704, −1.67761710559357250224591502339, 1.29142579399980925589510399729, 2.52660691696697297186030145230, 3.59487677645345835702357555328, 5.05644454065268951966455918329, 5.93037059274376646282672121959, 6.71889673871260167780627380936, 7.44911261909927954324449324381, 8.290197703494024700575096069023, 9.525693704969977313442153321542, 10.27683224942201915637788718752

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