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Properties

Label 2-2856-2856.1427-c0-0-11
Degree $2$
Conductor $2856$
Sign $1$
Analytic cond. $1.42532$
Root an. cond. $1.19387$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s + 5-s − 6-s − 7-s − 8-s + 9-s − 10-s − 11-s + 12-s + 13-s + 14-s + 15-s + 16-s + 17-s − 18-s + 20-s − 21-s + 22-s − 24-s − 26-s + 27-s − 28-s − 30-s − 32-s − 33-s + ⋯
L(s)  = 1  − 2-s + 3-s + 4-s + 5-s − 6-s − 7-s − 8-s + 9-s − 10-s − 11-s + 12-s + 13-s + 14-s + 15-s + 16-s + 17-s − 18-s + 20-s − 21-s + 22-s − 24-s − 26-s + 27-s − 28-s − 30-s − 32-s − 33-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2856\)    =    \(2^{3} \cdot 3 \cdot 7 \cdot 17\)
Sign: $1$
Analytic conductor: \(1.42532\)
Root analytic conductor: \(1.19387\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2856} (1427, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2856,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.282141484\)
\(L(\frac12)\) \(\approx\) \(1.282141484\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−8.966209306155156683366282574306, −8.368601849865015168711863454411, −7.63921353472406810671670450749, −6.87258168255748518065606283844, −6.07764231773660794551496109383, −5.43532887251237540194712724526, −3.82739914979174261907930391675, −2.98961212124006038709005470793, −2.33142951080950305408421192783, −1.23954574406193814077301980605, 1.23954574406193814077301980605, 2.33142951080950305408421192783, 2.98961212124006038709005470793, 3.82739914979174261907930391675, 5.43532887251237540194712724526, 6.07764231773660794551496109383, 6.87258168255748518065606283844, 7.63921353472406810671670450749, 8.368601849865015168711863454411, 8.966209306155156683366282574306

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