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Properties

Label 2-2768-692.691-c0-0-2
Degree $2$
Conductor $2768$
Sign $1$
Analytic cond. $1.38141$
Root an. cond. $1.17533$
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  − 9-s + 2·13-s + 25-s − 2·29-s + 2·37-s + 2·41-s − 49-s + 2·73-s + 81-s + 2·89-s − 2·109-s − 2·113-s − 2·117-s + ⋯
L(s)  = 1  − 9-s + 2·13-s + 25-s − 2·29-s + 2·37-s + 2·41-s − 49-s + 2·73-s + 81-s + 2·89-s − 2·109-s − 2·113-s − 2·117-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2768\)    =    \(2^{4} \cdot 173\)
Sign: $1$
Analytic conductor: \(1.38141\)
Root analytic conductor: \(1.17533\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2768} (2767, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2768,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.243977979\)
\(L(\frac12)\) \(\approx\) \(1.243977979\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
173 \( 1 + T \)
good3 \( 1 + T^{2} \)
5 \( ( 1 - T )( 1 + T ) \)
7 \( 1 + T^{2} \)
11 \( 1 + T^{2} \)
13 \( ( 1 - T )^{2} \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( 1 + T^{2} \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 + T )^{2} \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( ( 1 - T )^{2} \)
41 \( ( 1 - T )^{2} \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( 1 + T^{2} \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( 1 + T^{2} \)
73 \( ( 1 - T )^{2} \)
79 \( 1 + T^{2} \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 - T )^{2} \)
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

−9.095529352724138277330811996063, −8.203168186995837253958973435111, −7.69874786240718133520742752717, −6.48400449310595035557330147204, −6.00762537148756171712337890928, −5.27170032100690686198140636077, −4.10816457031332010557321626667, −3.40658589331748079084676392612, −2.41497110594417284475174556281, −1.08201986441596680799131560523, 1.08201986441596680799131560523, 2.41497110594417284475174556281, 3.40658589331748079084676392612, 4.10816457031332010557321626667, 5.27170032100690686198140636077, 6.00762537148756171712337890928, 6.48400449310595035557330147204, 7.69874786240718133520742752717, 8.203168186995837253958973435111, 9.095529352724138277330811996063

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