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Properties

Label 2-2856-2856.2099-c0-0-2
Degree $2$
Conductor $2856$
Sign $-0.892 + 0.451i$
Analytic cond. $1.42532$
Root an. cond. $1.19387$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 0.707i)2-s + (0.382 − 0.923i)3-s − 1.00i·4-s + (0.382 + 0.923i)6-s + (−0.923 + 0.382i)7-s + (0.707 + 0.707i)8-s + (−0.707 − 0.707i)9-s + (−0.923 − 0.382i)12-s + 0.765i·13-s + (0.382 − 0.923i)14-s − 1.00·16-s + (−0.923 − 0.382i)17-s + 18-s + i·21-s + (−0.707 − 1.70i)23-s + (0.923 − 0.382i)24-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (0.382 − 0.923i)3-s − 1.00i·4-s + (0.382 + 0.923i)6-s + (−0.923 + 0.382i)7-s + (0.707 + 0.707i)8-s + (−0.707 − 0.707i)9-s + (−0.923 − 0.382i)12-s + 0.765i·13-s + (0.382 − 0.923i)14-s − 1.00·16-s + (−0.923 − 0.382i)17-s + 18-s + i·21-s + (−0.707 − 1.70i)23-s + (0.923 − 0.382i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.892 + 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.892 + 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2856\)    =    \(2^{3} \cdot 3 \cdot 7 \cdot 17\)
Sign: $-0.892 + 0.451i$
Analytic conductor: \(1.42532\)
Root analytic conductor: \(1.19387\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2856} (2099, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2856,\ (\ :0),\ -0.892 + 0.451i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2370986198\)
\(L(\frac12)\) \(\approx\) \(0.2370986198\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (-0.382 + 0.923i)T \)
7 \( 1 + (0.923 - 0.382i)T \)
17 \( 1 + (0.923 + 0.382i)T \)
good5 \( 1 + (-0.707 - 0.707i)T^{2} \)
11 \( 1 + (0.707 - 0.707i)T^{2} \)
13 \( 1 - 0.765iT - T^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \)
29 \( 1 + (1.70 + 0.707i)T + (0.707 + 0.707i)T^{2} \)
31 \( 1 + (-0.707 - 0.707i)T^{2} \)
37 \( 1 + (0.707 + 0.707i)T^{2} \)
41 \( 1 + (1.30 - 0.541i)T + (0.707 - 0.707i)T^{2} \)
43 \( 1 + (1 + i)T + iT^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (-1 + i)T - iT^{2} \)
59 \( 1 + (0.541 + 0.541i)T + iT^{2} \)
61 \( 1 + (1.30 - 0.541i)T + (0.707 - 0.707i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (0.292 - 0.707i)T + (-0.707 - 0.707i)T^{2} \)
73 \( 1 + (-0.707 - 0.707i)T^{2} \)
79 \( 1 + (0.707 - 0.707i)T^{2} \)
83 \( 1 + (-1.30 + 1.30i)T - iT^{2} \)
89 \( 1 - 1.84iT - T^{2} \)
97 \( 1 + (-0.707 - 0.707i)T^{2} \)
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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

−8.724824862249093637319604385124, −7.900251602296295826798363205859, −6.95698897021102209088429287555, −6.66396205421307157446586117809, −5.98027020492223284093535768589, −5.01792094143780346947085768916, −3.80786688147358463372567366899, −2.54302617836597129746027144153, −1.80477123991750153532770393337, −0.16613707672791494726918440013, 1.77216219288162323996382783735, 2.95078405322451662265534037129, 3.52231714927019638464615089448, 4.22642601500096466954017249882, 5.28566245961116803951076121048, 6.31276178619241689983352595545, 7.33341210579778554280133155971, 7.956460661109213163792829779821, 8.857785958188969456787295653136, 9.304643269072489393522053038469

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