Properties

Label 2-6762-1.1-c1-0-78
Degree $2$
Conductor $6762$
Sign $-1$
Analytic cond. $53.9948$
Root an. cond. $7.34811$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 4.26·5-s − 6-s + 8-s + 9-s − 4.26·10-s − 3.76·11-s − 12-s + 2.26·13-s + 4.26·15-s + 16-s − 0.352·17-s + 18-s − 2·19-s − 4.26·20-s − 3.76·22-s − 23-s − 24-s + 13.1·25-s + 2.26·26-s − 27-s + 9.13·29-s + 4.26·30-s + 7.89·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.90·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s − 1.34·10-s − 1.13·11-s − 0.288·12-s + 0.626·13-s + 1.10·15-s + 0.250·16-s − 0.0855·17-s + 0.235·18-s − 0.458·19-s − 0.952·20-s − 0.803·22-s − 0.208·23-s − 0.204·24-s + 2.63·25-s + 0.443·26-s − 0.192·27-s + 1.69·29-s + 0.777·30-s + 1.41·31-s + 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6762\)    =    \(2 \cdot 3 \cdot 7^{2} \cdot 23\)
Sign: $-1$
Analytic conductor: \(53.9948\)
Root analytic conductor: \(7.34811\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{6762} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6762,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + T \)
7 \( 1 \)
23 \( 1 + T \)
good5 \( 1 + 4.26T + 5T^{2} \)
11 \( 1 + 3.76T + 11T^{2} \)
13 \( 1 - 2.26T + 13T^{2} \)
17 \( 1 + 0.352T + 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
29 \( 1 - 9.13T + 29T^{2} \)
31 \( 1 - 7.89T + 31T^{2} \)
37 \( 1 - 3.15T + 37T^{2} \)
41 \( 1 + 5.90T + 41T^{2} \)
43 \( 1 - 9.67T + 43T^{2} \)
47 \( 1 + 9.27T + 47T^{2} \)
53 \( 1 + 10.2T + 53T^{2} \)
59 \( 1 - 9.74T + 59T^{2} \)
61 \( 1 + 1.12T + 61T^{2} \)
67 \( 1 + 8.99T + 67T^{2} \)
71 \( 1 - 3.29T + 71T^{2} \)
73 \( 1 + 3.01T + 73T^{2} \)
79 \( 1 - 3.01T + 79T^{2} \)
83 \( 1 + 7.22T + 83T^{2} \)
89 \( 1 + 4.35T + 89T^{2} \)
97 \( 1 + 16.8T + 97T^{2} \)
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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

−7.64422508429803792453256680746, −6.80133788865655600497300030420, −6.31274584166073460430288631475, −5.31729564532094992504864233480, −4.55622416418944421976212204810, −4.25559087899004267513693077360, −3.27287710794807709521989652806, −2.66399943960803245393479694401, −1.08795194773697262277374519416, 0, 1.08795194773697262277374519416, 2.66399943960803245393479694401, 3.27287710794807709521989652806, 4.25559087899004267513693077360, 4.55622416418944421976212204810, 5.31729564532094992504864233480, 6.31274584166073460430288631475, 6.80133788865655600497300030420, 7.64422508429803792453256680746

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