Properties

Label 2-6762-1.1-c1-0-96
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 + 2.70·5-s + 6-s − 8-s + 9-s − 2.70·10-s − 4·11-s − 12-s + 0.701·13-s − 2.70·15-s + 16-s − 4·17-s − 18-s − 7.40·19-s + 2.70·20-s + 4·22-s + 23-s + 24-s + 2.29·25-s − 0.701·26-s − 27-s + 6.70·29-s + 2.70·30-s + 2·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.20·5-s + 0.408·6-s − 0.353·8-s + 0.333·9-s − 0.854·10-s − 1.20·11-s − 0.288·12-s + 0.194·13-s − 0.697·15-s + 0.250·16-s − 0.970·17-s − 0.235·18-s − 1.69·19-s + 0.604·20-s + 0.852·22-s + 0.208·23-s + 0.204·24-s + 0.459·25-s − 0.137·26-s − 0.192·27-s + 1.24·29-s + 0.493·30-s + 0.359·31-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 \( 1 - 2.70T + 5T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 - 0.701T + 13T^{2} \)
17 \( 1 + 4T + 17T^{2} \)
19 \( 1 + 7.40T + 19T^{2} \)
29 \( 1 - 6.70T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 - 10.7T + 37T^{2} \)
41 \( 1 - 6.70T + 41T^{2} \)
43 \( 1 - 4.70T + 43T^{2} \)
47 \( 1 + 8.10T + 47T^{2} \)
53 \( 1 - 3.40T + 53T^{2} \)
59 \( 1 + 5.40T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + 10.8T + 67T^{2} \)
71 \( 1 - 5.40T + 71T^{2} \)
73 \( 1 - 11.4T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 0.596T + 83T^{2} \)
89 \( 1 + 1.40T + 89T^{2} \)
97 \( 1 + 12.7T + 97T^{2} \)
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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

−7.78374039803056452565452130652, −6.66483433370606096590708027723, −6.36001638344995593006415324186, −5.71563500775065685159917822189, −4.88635039386829524745743437402, −4.17657458629832033977786561339, −2.61019168132861997162538230383, −2.34116508317957948561264588235, −1.20268176913359249348880177949, 0, 1.20268176913359249348880177949, 2.34116508317957948561264588235, 2.61019168132861997162538230383, 4.17657458629832033977786561339, 4.88635039386829524745743437402, 5.71563500775065685159917822189, 6.36001638344995593006415324186, 6.66483433370606096590708027723, 7.78374039803056452565452130652

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