Properties

Label 2-6762-1.1-c1-0-36
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 − 3.70·5-s + 6-s − 8-s + 9-s + 3.70·10-s − 4·11-s − 12-s − 5.70·13-s + 3.70·15-s + 16-s − 4·17-s − 18-s + 5.40·19-s − 3.70·20-s + 4·22-s + 23-s + 24-s + 8.70·25-s + 5.70·26-s − 27-s + 0.298·29-s − 3.70·30-s + 2·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.65·5-s + 0.408·6-s − 0.353·8-s + 0.333·9-s + 1.17·10-s − 1.20·11-s − 0.288·12-s − 1.58·13-s + 0.955·15-s + 0.250·16-s − 0.970·17-s − 0.235·18-s + 1.23·19-s − 0.827·20-s + 0.852·22-s + 0.208·23-s + 0.204·24-s + 1.74·25-s + 1.11·26-s − 0.192·27-s + 0.0554·29-s − 0.675·30-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 \( 1 + 3.70T + 5T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 + 5.70T + 13T^{2} \)
17 \( 1 + 4T + 17T^{2} \)
19 \( 1 - 5.40T + 19T^{2} \)
29 \( 1 - 0.298T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 - 4.29T + 37T^{2} \)
41 \( 1 - 0.298T + 41T^{2} \)
43 \( 1 + 1.70T + 43T^{2} \)
47 \( 1 - 11.1T + 47T^{2} \)
53 \( 1 + 9.40T + 53T^{2} \)
59 \( 1 - 7.40T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 - 14.8T + 67T^{2} \)
71 \( 1 + 7.40T + 71T^{2} \)
73 \( 1 + 1.40T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 13.4T + 83T^{2} \)
89 \( 1 - 11.4T + 89T^{2} \)
97 \( 1 + 6.29T + 97T^{2} \)
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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

−7.49191662657071870712973170466, −7.33405043483349563643895621019, −6.48272235791769674930536957301, −5.33810153645552896380643486623, −4.84692657850520050187422770260, −4.06563861416579367207014785127, −3.02708975877617637456645976540, −2.34131916182092945441788993781, −0.77228464635059305371408620502, 0, 0.77228464635059305371408620502, 2.34131916182092945441788993781, 3.02708975877617637456645976540, 4.06563861416579367207014785127, 4.84692657850520050187422770260, 5.33810153645552896380643486623, 6.48272235791769674930536957301, 7.33405043483349563643895621019, 7.49191662657071870712973170466

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