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Properties

Label 2-1856-116.115-c0-0-3
Degree $2$
Conductor $1856$
Sign $1$
Analytic cond. $0.926264$
Root an. cond. $0.962426$
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  + 3-s + 5-s + 11-s + 13-s + 15-s − 2·19-s − 27-s − 29-s − 31-s + 33-s + 39-s + 43-s − 47-s + 49-s + 53-s + 55-s − 2·57-s + 65-s − 79-s − 81-s − 87-s − 93-s − 2·95-s + 109-s + ⋯
L(s)  = 1  + 3-s + 5-s + 11-s + 13-s + 15-s − 2·19-s − 27-s − 29-s − 31-s + 33-s + 39-s + 43-s − 47-s + 49-s + 53-s + 55-s − 2·57-s + 65-s − 79-s − 81-s − 87-s − 93-s − 2·95-s + 109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1856\)    =    \(2^{6} \cdot 29\)
Sign: $1$
Analytic conductor: \(0.926264\)
Root analytic conductor: \(0.962426\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1856} (1855, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1856,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.856441814\)
\(L(\frac12)\) \(\approx\) \(1.856441814\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
29 \( 1 + T \)
good3 \( 1 - T + T^{2} \)
5 \( 1 - T + T^{2} \)
7 \( ( 1 - T )( 1 + T ) \)
11 \( 1 - T + T^{2} \)
13 \( 1 - T + T^{2} \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( ( 1 + T )^{2} \)
23 \( ( 1 - T )( 1 + T ) \)
31 \( 1 + T + T^{2} \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( 1 - T + T^{2} \)
47 \( 1 + T + T^{2} \)
53 \( 1 - T + T^{2} \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( 1 + T + T^{2} \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( ( 1 - T )( 1 + T ) \)
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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

−9.086086375193338685405200180370, −8.915677501785756212802516359914, −8.097064021907919610822594671392, −7.04756105550923081864774535258, −6.15973544484094960456087000549, −5.66388429240353944279540536859, −4.20451322010249716076597518961, −3.58950748124436568628292357784, −2.35663345469056819731409571948, −1.67615022476033855220625824411, 1.67615022476033855220625824411, 2.35663345469056819731409571948, 3.58950748124436568628292357784, 4.20451322010249716076597518961, 5.66388429240353944279540536859, 6.15973544484094960456087000549, 7.04756105550923081864774535258, 8.097064021907919610822594671392, 8.915677501785756212802516359914, 9.086086375193338685405200180370

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