Properties

Label 2-6762-1.1-c1-0-113
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 − 2·5-s − 6-s + 8-s + 9-s − 2·10-s − 12-s + 4.47·13-s + 2·15-s + 16-s − 4.47·17-s + 18-s − 2.47·19-s − 2·20-s − 23-s − 24-s − 25-s + 4.47·26-s − 27-s − 2·29-s + 2·30-s + 2.47·31-s + 32-s − 4.47·34-s + 36-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.894·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s − 0.632·10-s − 0.288·12-s + 1.24·13-s + 0.516·15-s + 0.250·16-s − 1.08·17-s + 0.235·18-s − 0.567·19-s − 0.447·20-s − 0.208·23-s − 0.204·24-s − 0.200·25-s + 0.877·26-s − 0.192·27-s − 0.371·29-s + 0.365·30-s + 0.444·31-s + 0.176·32-s − 0.766·34-s + 0.166·36-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 \( 1 + 2T + 5T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 4.47T + 13T^{2} \)
17 \( 1 + 4.47T + 17T^{2} \)
19 \( 1 + 2.47T + 19T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 - 2.47T + 31T^{2} \)
37 \( 1 - 6.94T + 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + 4.94T + 43T^{2} \)
47 \( 1 - 2.47T + 47T^{2} \)
53 \( 1 + 10.9T + 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 - 4.94T + 67T^{2} \)
71 \( 1 - 4.94T + 71T^{2} \)
73 \( 1 + 14.9T + 73T^{2} \)
79 \( 1 + 4.94T + 79T^{2} \)
83 \( 1 + 2.47T + 83T^{2} \)
89 \( 1 + 9.41T + 89T^{2} \)
97 \( 1 - 0.472T + 97T^{2} \)
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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

−7.56949522208829419460220157969, −6.68561576903558979322592974762, −6.23024237043829594777686184890, −5.55498928191957357784180483372, −4.51389790047837359627473796013, −4.18153104469561100531937631122, −3.43440665376120551085979052233, −2.39710679754217439154470131666, −1.29491499148409965249544355272, 0, 1.29491499148409965249544355272, 2.39710679754217439154470131666, 3.43440665376120551085979052233, 4.18153104469561100531937631122, 4.51389790047837359627473796013, 5.55498928191957357784180483372, 6.23024237043829594777686184890, 6.68561576903558979322592974762, 7.56949522208829419460220157969

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