Properties

Label 2-768-1.1-c3-0-23
Degree $2$
Conductor $768$
Sign $-1$
Analytic cond. $45.3134$
Root an. cond. $6.73152$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 18.5·5-s + 9.32·7-s + 9·9-s − 39.7·11-s + 32.9·13-s + 55.6·15-s + 90.5·17-s + 72.5·19-s − 27.9·21-s + 45.3·23-s + 218.·25-s − 27·27-s − 143.·29-s − 90.4·31-s + 119.·33-s − 172.·35-s − 1.77·37-s − 98.8·39-s − 195.·41-s + 407.·43-s − 166.·45-s + 278.·47-s − 256.·49-s − 271.·51-s − 241.·53-s + 736.·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.65·5-s + 0.503·7-s + 0.333·9-s − 1.08·11-s + 0.703·13-s + 0.957·15-s + 1.29·17-s + 0.876·19-s − 0.290·21-s + 0.411·23-s + 1.75·25-s − 0.192·27-s − 0.918·29-s − 0.524·31-s + 0.628·33-s − 0.835·35-s − 0.00790·37-s − 0.405·39-s − 0.745·41-s + 1.44·43-s − 0.552·45-s + 0.864·47-s − 0.746·49-s − 0.746·51-s − 0.625·53-s + 1.80·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $-1$
Analytic conductor: \(45.3134\)
Root analytic conductor: \(6.73152\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 768,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + 3T \)
good5 \( 1 + 18.5T + 125T^{2} \)
7 \( 1 - 9.32T + 343T^{2} \)
11 \( 1 + 39.7T + 1.33e3T^{2} \)
13 \( 1 - 32.9T + 2.19e3T^{2} \)
17 \( 1 - 90.5T + 4.91e3T^{2} \)
19 \( 1 - 72.5T + 6.85e3T^{2} \)
23 \( 1 - 45.3T + 1.21e4T^{2} \)
29 \( 1 + 143.T + 2.43e4T^{2} \)
31 \( 1 + 90.4T + 2.97e4T^{2} \)
37 \( 1 + 1.77T + 5.06e4T^{2} \)
41 \( 1 + 195.T + 6.89e4T^{2} \)
43 \( 1 - 407.T + 7.95e4T^{2} \)
47 \( 1 - 278.T + 1.03e5T^{2} \)
53 \( 1 + 241.T + 1.48e5T^{2} \)
59 \( 1 + 149.T + 2.05e5T^{2} \)
61 \( 1 + 508.T + 2.26e5T^{2} \)
67 \( 1 - 950.T + 3.00e5T^{2} \)
71 \( 1 + 803.T + 3.57e5T^{2} \)
73 \( 1 + 449.T + 3.89e5T^{2} \)
79 \( 1 - 157.T + 4.93e5T^{2} \)
83 \( 1 + 175.T + 5.71e5T^{2} \)
89 \( 1 + 127.T + 7.04e5T^{2} \)
97 \( 1 - 158.T + 9.12e5T^{2} \)
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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

−9.545926400828767969814625644481, −8.361679690073748069119438934250, −7.70353582395919625639268348945, −7.21522431960868665290002735993, −5.75003461029218146942210396831, −5.00723955911273394122720965364, −3.96262756695492195544689232680, −3.08474330877103364898518681223, −1.18864037236491074372327602677, 0, 1.18864037236491074372327602677, 3.08474330877103364898518681223, 3.96262756695492195544689232680, 5.00723955911273394122720965364, 5.75003461029218146942210396831, 7.21522431960868665290002735993, 7.70353582395919625639268348945, 8.361679690073748069119438934250, 9.545926400828767969814625644481

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