Properties

Label 2-768-1.1-c1-0-13
Degree $2$
Conductor $768$
Sign $-1$
Analytic cond. $6.13251$
Root an. cond. $2.47639$
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  − 3-s + 2.82·5-s − 2.82·7-s + 9-s − 4·11-s − 5.65·13-s − 2.82·15-s − 2·17-s − 4·19-s + 2.82·21-s + 5.65·23-s + 3.00·25-s − 27-s − 2.82·29-s + 8.48·31-s + 4·33-s − 8.00·35-s + 5.65·39-s − 10·41-s − 12·43-s + 2.82·45-s − 5.65·47-s + 1.00·49-s + 2·51-s − 2.82·53-s − 11.3·55-s + 4·57-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.26·5-s − 1.06·7-s + 0.333·9-s − 1.20·11-s − 1.56·13-s − 0.730·15-s − 0.485·17-s − 0.917·19-s + 0.617·21-s + 1.17·23-s + 0.600·25-s − 0.192·27-s − 0.525·29-s + 1.52·31-s + 0.696·33-s − 1.35·35-s + 0.905·39-s − 1.56·41-s − 1.82·43-s + 0.421·45-s − 0.825·47-s + 0.142·49-s + 0.280·51-s − 0.388·53-s − 1.52·55-s + 0.529·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $-1$
Analytic conductor: \(6.13251\)
Root analytic conductor: \(2.47639\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 768,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 + T \)
good5 \( 1 - 2.82T + 5T^{2} \)
7 \( 1 + 2.82T + 7T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 + 5.65T + 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 - 5.65T + 23T^{2} \)
29 \( 1 + 2.82T + 29T^{2} \)
31 \( 1 - 8.48T + 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 + 10T + 41T^{2} \)
43 \( 1 + 12T + 43T^{2} \)
47 \( 1 + 5.65T + 47T^{2} \)
53 \( 1 + 2.82T + 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 - 11.3T + 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + 5.65T + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 - 8.48T + 79T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 - 14T + 97T^{2} \)
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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

−10.07245506107097851006983775266, −9.365635563042269871264855048440, −8.214583816897408615776573430204, −6.91079549811893204270195542090, −6.46934467573382852912407027576, −5.35749205002680040951818323440, −4.79742178821599754690515469672, −3.02446976059117841138850052879, −2.10278277956094620876076270023, 0, 2.10278277956094620876076270023, 3.02446976059117841138850052879, 4.79742178821599754690515469672, 5.35749205002680040951818323440, 6.46934467573382852912407027576, 6.91079549811893204270195542090, 8.214583816897408615776573430204, 9.365635563042269871264855048440, 10.07245506107097851006983775266

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