Properties

Label 2-6762-1.1-c1-0-97
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 − 3.41·5-s − 6-s + 8-s + 9-s − 3.41·10-s + 2.29·11-s − 12-s + 1.54·13-s + 3.41·15-s + 16-s + 2.72·17-s + 18-s − 7.15·19-s − 3.41·20-s + 2.29·22-s + 23-s − 24-s + 6.65·25-s + 1.54·26-s − 27-s − 9.69·29-s + 3.41·30-s − 5.15·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.52·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s − 1.07·10-s + 0.693·11-s − 0.288·12-s + 0.429·13-s + 0.881·15-s + 0.250·16-s + 0.659·17-s + 0.235·18-s − 1.64·19-s − 0.763·20-s + 0.490·22-s + 0.208·23-s − 0.204·24-s + 1.33·25-s + 0.303·26-s − 0.192·27-s − 1.80·29-s + 0.623·30-s − 0.925·31-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 \( 1 + 3.41T + 5T^{2} \)
11 \( 1 - 2.29T + 11T^{2} \)
13 \( 1 - 1.54T + 13T^{2} \)
17 \( 1 - 2.72T + 17T^{2} \)
19 \( 1 + 7.15T + 19T^{2} \)
29 \( 1 + 9.69T + 29T^{2} \)
31 \( 1 + 5.15T + 31T^{2} \)
37 \( 1 - 2.72T + 37T^{2} \)
41 \( 1 - 6.01T + 41T^{2} \)
43 \( 1 - 11.8T + 43T^{2} \)
47 \( 1 - 0.557T + 47T^{2} \)
53 \( 1 + 3.31T + 53T^{2} \)
59 \( 1 - 7.69T + 59T^{2} \)
61 \( 1 - 0.0113T + 61T^{2} \)
67 \( 1 - 8.93T + 67T^{2} \)
71 \( 1 + 9.25T + 71T^{2} \)
73 \( 1 - 8.09T + 73T^{2} \)
79 \( 1 + 10.4T + 79T^{2} \)
83 \( 1 + 1.71T + 83T^{2} \)
89 \( 1 + 16.0T + 89T^{2} \)
97 \( 1 - 5.76T + 97T^{2} \)
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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

−7.43502720658320660550084471721, −6.97973045184151956724217105069, −6.08822958838618054949392688811, −5.56832530938695419572344297698, −4.53803835243252435625748274658, −3.94865612432082169104247467268, −3.65751830233570174152884135918, −2.41701903562882615292488562165, −1.21437916918972185492028579897, 0, 1.21437916918972185492028579897, 2.41701903562882615292488562165, 3.65751830233570174152884135918, 3.94865612432082169104247467268, 4.53803835243252435625748274658, 5.56832530938695419572344297698, 6.08822958838618054949392688811, 6.97973045184151956724217105069, 7.43502720658320660550084471721

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