Properties

Label 2-6762-1.1-c1-0-98
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 − 5-s − 6-s − 8-s + 9-s + 10-s − 0.404·11-s + 12-s − 4.12·13-s − 15-s + 16-s − 5.31·17-s − 18-s + 2.71·19-s − 20-s + 0.404·22-s − 23-s − 24-s − 4·25-s + 4.12·26-s + 27-s + 9.12·29-s + 30-s + 9.83·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.447·5-s − 0.408·6-s − 0.353·8-s + 0.333·9-s + 0.316·10-s − 0.121·11-s + 0.288·12-s − 1.14·13-s − 0.258·15-s + 0.250·16-s − 1.28·17-s − 0.235·18-s + 0.623·19-s − 0.223·20-s + 0.0862·22-s − 0.208·23-s − 0.204·24-s − 0.800·25-s + 0.808·26-s + 0.192·27-s + 1.69·29-s + 0.182·30-s + 1.76·31-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 \( 1 + T + 5T^{2} \)
11 \( 1 + 0.404T + 11T^{2} \)
13 \( 1 + 4.12T + 13T^{2} \)
17 \( 1 + 5.31T + 17T^{2} \)
19 \( 1 - 2.71T + 19T^{2} \)
29 \( 1 - 9.12T + 29T^{2} \)
31 \( 1 - 9.83T + 31T^{2} \)
37 \( 1 + 7.90T + 37T^{2} \)
41 \( 1 - 10.7T + 41T^{2} \)
43 \( 1 - 0.808T + 43T^{2} \)
47 \( 1 - 10.8T + 47T^{2} \)
53 \( 1 + 0.191T + 53T^{2} \)
59 \( 1 - 8.31T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 - 2.83T + 67T^{2} \)
71 \( 1 + 6.92T + 71T^{2} \)
73 \( 1 + 4.59T + 73T^{2} \)
79 \( 1 + 9.92T + 79T^{2} \)
83 \( 1 - 4.40T + 83T^{2} \)
89 \( 1 + 7.90T + 89T^{2} \)
97 \( 1 + 15.3T + 97T^{2} \)
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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

−7.77406838566217804838595446354, −7.08497648819684300969455536288, −6.54299172064182605358818132443, −5.55425206344771554238505584651, −4.57096024048729199789663386827, −4.04187144817444009989973873588, −2.74506833880287681186159289538, −2.50274142739295888620201324250, −1.21639286949392565591633363546, 0, 1.21639286949392565591633363546, 2.50274142739295888620201324250, 2.74506833880287681186159289538, 4.04187144817444009989973873588, 4.57096024048729199789663386827, 5.55425206344771554238505584651, 6.54299172064182605358818132443, 7.08497648819684300969455536288, 7.77406838566217804838595446354

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