Properties

Label 2-6762-1.1-c1-0-2
Degree $2$
Conductor $6762$
Sign $1$
Analytic cond. $53.9948$
Root an. cond. $7.34811$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 4·5-s − 6-s + 8-s + 9-s − 4·10-s − 5·11-s − 12-s − 2·13-s + 4·15-s + 16-s − 3·17-s + 18-s − 6·19-s − 4·20-s − 5·22-s + 23-s − 24-s + 11·25-s − 2·26-s − 27-s + 29-s + 4·30-s − 10·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.78·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 1.26·10-s − 1.50·11-s − 0.288·12-s − 0.554·13-s + 1.03·15-s + 1/4·16-s − 0.727·17-s + 0.235·18-s − 1.37·19-s − 0.894·20-s − 1.06·22-s + 0.208·23-s − 0.204·24-s + 11/5·25-s − 0.392·26-s − 0.192·27-s + 0.185·29-s + 0.730·30-s − 1.79·31-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6762,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4370221680\)
\(L(\frac12)\) \(\approx\) \(0.4370221680\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 7 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−7.67339266025786875869742980640, −7.35765822385844184183312871496, −6.62804001799292469414052768124, −5.73266263982238899039043082686, −4.89293431281830942284828383894, −4.51497805893048147062331691897, −3.75080384949105123045635417543, −2.94854830042915404680973251413, −2.00526065889502327456672416030, −0.29226961440845967199617384465, 0.29226961440845967199617384465, 2.00526065889502327456672416030, 2.94854830042915404680973251413, 3.75080384949105123045635417543, 4.51497805893048147062331691897, 4.89293431281830942284828383894, 5.73266263982238899039043082686, 6.62804001799292469414052768124, 7.35765822385844184183312871496, 7.67339266025786875869742980640

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